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This is an overview table with the 95 benchmark entries showing the TeX of the formula and if the translation to semantic LaTeX and Mathematica (CAS) was correct. If the translation was wrong, a more detailed explanation and categorization of the error is given. This table does not contain the actual translation results. You can reach this information by clicking the link in the most left column, e.g., [[Gold 1]]. | |||
We provide the entire benchmark as a JSON file too: [[MediaWiki:Gold-data.json|gold-data.json]]. | |||
{| class="wikitable sortable" | {| class="wikitable sortable" | ||
| Line 5: | Line 7: | ||
! colspan="3" | Entry Info | ! colspan="3" | Entry Info | ||
! colspan="2" | Translations | ! colspan="2" | Translations | ||
! colspan="5" | Reason For Failure | ! colspan="5" | Reason For Failure | ||
! colspan="2" | | ! colspan="2" | | ||
|- | |- | ||
! # !! Formula !! Title !! Semantic LaTeX !! CAS Translations !! Definition / Substitution !! Pattern Matching !! Derivatives / Primes !! Missing Infos !! Untranslatable Macro !! Explanation !! Evaluation Data | ! # !! Formula !! Title !! Semantic LaTeX !! CAS Translations !! Definition / Substitution !! Pattern Matching !! Derivatives / Primes !! Missing Infos !! Untranslatable Macro !! Explanation !! Evaluation Data | ||
|- | |- | ||
| 1 | | [[Gold 1]] | ||
| <syntaxhighlight lang="tex" inline >\begin{align}J_{-(m+\frac{1}{2})}(x) &= (-1)^{m+1} Y_{m+\frac{1}{2}}(x), \\Y_{-(m+\frac{1}{2})}(x) &= (-1)^m J_{m+\frac{1}{2}}(x).\end{align}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\begin{align}J_{-(m+\frac{1}{2})}(x) &= (-1)^{m+1} Y_{m+\frac{1}{2}}(x), \\Y_{-(m+\frac{1}{2})}(x) &= (-1)^m J_{m+\frac{1}{2}}(x).\end{align}</syntaxhighlight> | ||
| [[Bessel function#math.51.18| Bessel function]] | | [[Bessel function#math.51.18| Bessel function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 261: | Line 263: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 2 | | [[Gold 2]] | ||
| <syntaxhighlight lang="tex" inline >E(e) \,=\, \int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >E(e) \,=\, \int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta</syntaxhighlight> | ||
| [[Ellipse#math.52.404| Ellipse]] | | [[Ellipse#math.52.404| Ellipse]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 388: | Line 390: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 3 | | [[Gold 3]] | ||
| <syntaxhighlight lang="tex" inline >F(x;k) = u</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >F(x;k) = u</syntaxhighlight> | ||
| [[Elliptic integral#math.53.6| Elliptic integral]] | | [[Elliptic integral#math.53.6| Elliptic integral]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 495: | Line 497: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 4 | | [[Gold 4]] | ||
| <syntaxhighlight lang="tex" inline >\frac{1}{\Gamma(z)} = \frac{i}{2\pi}\int_C (-t)^{-z}e^{-t}\,dt</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\frac{1}{\Gamma(z)} = \frac{i}{2\pi}\int_C (-t)^{-z}e^{-t}\,dt</syntaxhighlight> | ||
| [[Gamma function#math.54.195| Gamma function]] | | [[Gamma function#math.54.195| Gamma function]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
| Line 585: | Line 587: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 5 | | [[Gold 5]] | ||
| <syntaxhighlight lang="tex" inline >2^{4} = 2 \times2 \times 2 \times 2 = 16</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >2^{4} = 2 \times2 \times 2 \times 2 = 16</syntaxhighlight> | ||
| [[Logarithm#| Logarithm]] | | [[Logarithm#| Logarithm]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 661: | Line 663: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 6 | | [[Gold 6]] | ||
| <syntaxhighlight lang="tex" inline >\psi(x) := \sum_{n=1}^\infty e^{-n^2 \pi x}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\psi(x) := \sum_{n=1}^\infty e^{-n^2 \pi x}</syntaxhighlight> | ||
| [[Riemann zeta function#math.56.40| Riemann zeta function]] | | [[Riemann zeta function#math.56.40| Riemann zeta function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 746: | Line 748: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 7 | | [[Gold 7]] | ||
| <syntaxhighlight lang="tex" inline >\operatorname{li}(x) = \lim_{\varepsilon \to 0+} \left( \int_0^{1-\varepsilon} \frac{dt}{\ln t} + \int_{1+\varepsilon}^x \frac{dt}{\ln t} \right)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\operatorname{li}(x) = \lim_{\varepsilon \to 0+} \left( \int_0^{1-\varepsilon} \frac{dt}{\ln t} + \int_{1+\varepsilon}^x \frac{dt}{\ln t} \right)</syntaxhighlight> | ||
| [[Logarithmic integral function#math.57.2| Logarithmic integral function]] | | [[Logarithmic integral function#math.57.2| Logarithmic integral function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 858: | Line 860: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 8 | | [[Gold 8]] | ||
| <syntaxhighlight lang="tex" inline >w_{i} = \frac{1}{p'_{n}(x_{i})}\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >w_{i} = \frac{1}{p'_{n}(x_{i})}\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx</syntaxhighlight> | ||
| [[Gaussian quadrature#math.58.61| Gaussian quadrature]] | | [[Gaussian quadrature#math.58.61| Gaussian quadrature]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 955: | Line 957: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 9 | | [[Gold 9]] | ||
| <syntaxhighlight lang="tex" inline >\begin{align}x & =ue^u, \\[5pt]\frac{dx}{du} & =(u+1)e^u.\end{align}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\begin{align}x & =ue^u, \\[5pt]\frac{dx}{du} & =(u+1)e^u.\end{align}</syntaxhighlight> | ||
| [[Lambert W function#math.59.52| Lambert W function]] | | [[Lambert W function#math.59.52| Lambert W function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 1,045: | Line 1,047: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 10 | | [[Gold 10]] | ||
| <syntaxhighlight lang="tex" inline >\frac{1}{\left| \mathbf{x}-\mathbf{x}' \right|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} P_\ell(\cos \gamma)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\frac{1}{\left| \mathbf{x}-\mathbf{x}' \right|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} P_\ell(\cos \gamma)</syntaxhighlight> | ||
| [[Legendre polynomials#math.60.57| Legendre polynomials]] | | [[Legendre polynomials#math.60.57| Legendre polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 1,127: | Line 1,129: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 11 | | [[Gold 11]] | ||
| <syntaxhighlight lang="tex" inline >\operatorname{erf}^{(k)}(z) = \frac{2 (-1)^{k-1}}{\sqrt{\pi}} \mathit{H}_{k-1}(z) e^{-z^2} = \frac{2}{\sqrt{\pi}} \frac{d^{k-1}}{dz^{k-1}} \left(e^{-z^2}\right),\qquad k=1, 2, \dots</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\operatorname{erf}^{(k)}(z) = \frac{2 (-1)^{k-1}}{\sqrt{\pi}} \mathit{H}_{k-1}(z) e^{-z^2} = \frac{2}{\sqrt{\pi}} \frac{d^{k-1}}{dz^{k-1}} \left(e^{-z^2}\right),\qquad k=1, 2, \dots</syntaxhighlight> | ||
| [[Error function#math.61.27| Error function]] | | [[Error function#math.61.27| Error function]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
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| - | | - | ||
| - | | - | ||
| <math>\operatorname{erf}^(k)</math> was not detected as {{math|k}}-th derivative but as power. | | <math>\operatorname{erf}^{(k)}</math> was not detected as {{math|k}}-th derivative but as power. | ||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | | <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | ||
"id": 11, | "id": 11, | ||
| Line 1,247: | Line 1,249: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 12 | | [[Gold 12]] | ||
| <syntaxhighlight lang="tex" inline >x_k = \cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >x_k = \cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1</syntaxhighlight> | ||
| [[Chebyshev polynomials#math.62.44| Chebyshev polynomials]] | | [[Chebyshev polynomials#math.62.44| Chebyshev polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 1,363: | Line 1,365: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 13 | | [[Gold 13]] | ||
| <syntaxhighlight lang="tex" inline >E(x, y; u) := \sum_{n=0}^\infty u^n \, \psi_n (x) \, \psi_n (y) = \frac{1}{\sqrt{\pi (1 - u^2)}} \, \exp\left(-\frac{1 - u}{1 + u} \, \frac{(x + y)^2}{4} - \frac{1 + u}{1 - u} \, \frac{(x - y)^2}{4}\right)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >E(x, y; u) := \sum_{n=0}^\infty u^n \, \psi_n (x) \, \psi_n (y) = \frac{1}{\sqrt{\pi (1 - u^2)}} \, \exp\left(-\frac{1 - u}{1 + u} \, \frac{(x + y)^2}{4} - \frac{1 + u}{1 - u} \, \frac{(x - y)^2}{4}\right)</syntaxhighlight> | ||
| [[Hermite polynomials#math.63.109| Hermite polynomials]] | | [[Hermite polynomials#math.63.109| Hermite polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 1,460: | Line 1,462: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 14 | | [[Gold 14]] | ||
| <syntaxhighlight lang="tex" inline >x = \pm 1</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >x = \pm 1</syntaxhighlight> | ||
| [[Legendre function#math.64.8| Legendre function]] | | [[Legendre function#math.64.8| Legendre function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 1,537: | Line 1,539: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 15 | | [[Gold 15]] | ||
| <syntaxhighlight lang="tex" inline >E_n=2^nE_n(\tfrac{1}{2})</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >E_n=2^nE_n(\tfrac{1}{2})</syntaxhighlight> | ||
| [[Bernoulli polynomials#math.65.27| Bernoulli polynomials]] | | [[Bernoulli polynomials#math.65.27| Bernoulli polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 1,592: | Line 1,594: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 16 | | [[Gold 16]] | ||
| <syntaxhighlight lang="tex" inline >\operatorname{Si}(ix) = i\operatorname{Shi}(x)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\operatorname{Si}(ix) = i\operatorname{Shi}(x)</syntaxhighlight> | ||
| [[Trigonometric integral#math.66.8| Trigonometric integral]] | | [[Trigonometric integral#math.66.8| Trigonometric integral]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| - | | - | ||
| - | | - | ||
| - | | - | ||
| | | - | ||
| Integral was not tagged as a noun by CoreNLP. Hence, the macro for hyperbolic sine function was retrieved too late and not considered for replacements. | |||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | | <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | ||
"id": 16, | "id": 16, | ||
| Line 1,683: | Line 1,685: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 17 | | [[Gold 17]] | ||
| <syntaxhighlight lang="tex" inline >f(z)=\frac{1}{\Beta(x,y)}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >f(z)=\frac{1}{\Beta(x,y)}</syntaxhighlight> | ||
| [[Beta function#math.67.29| Beta function]] | | [[Beta function#math.67.29| Beta function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 1,767: | Line 1,769: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 18 | | [[Gold 18]] | ||
| <syntaxhighlight lang="tex" inline >\begin{align}\int x^m e^{ix^n}\,dx & =\frac{x^{m+1}}{m+1}\,_1F_1\left(\begin{array}{c} \frac{m+1}{n}\\1+\frac{m+1}{n}\end{array}\mid ix^n\right) \\[6px]& =\frac{1}{n} i^\frac{m+1}{n}\gamma\left(\frac{m+1}{n},-ix^n\right),\end{align}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\begin{align}\int x^m e^{ix^n}\,dx & =\frac{x^{m+1}}{m+1}\,_1F_1\left(\begin{array}{c} \frac{m+1}{n}\\1+\frac{m+1}{n}\end{array}\mid ix^n\right) \\[6px]& =\frac{1}{n} i^\frac{m+1}{n}\gamma\left(\frac{m+1}{n},-ix^n\right),\end{align}</syntaxhighlight> | ||
| [[Fresnel integral#math.68.51| Fresnel integral]] | | [[Fresnel integral#math.68.51| Fresnel integral]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 1,833: | Line 1,835: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 19 | | [[Gold 19]] | ||
| <syntaxhighlight lang="tex" inline >T_n(x) = \frac{\Gamma(1/2)\sqrt{1-x^2}}{(-2)^n\,\Gamma(n+1/2)} \ \frac{d^n}{dx^n}\left([1-x^2]^{n-1/2}\right)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >T_n(x) = \frac{\Gamma(1/2)\sqrt{1-x^2}}{(-2)^n\,\Gamma(n+1/2)} \ \frac{d^n}{dx^n}\left([1-x^2]^{n-1/2}\right)</syntaxhighlight> | ||
| [[Classical orthogonal polynomials#math.69.117| Classical orthogonal polynomials]] | | [[Classical orthogonal polynomials#math.69.117| Classical orthogonal polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 1,918: | Line 1,920: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 20 | | [[Gold 20]] | ||
| <syntaxhighlight lang="tex" inline >{}_1F_0(1;;z) = \sum_{n \geqslant 0} z^n = (1-z)^{-1}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >{}_1F_0(1;;z) = \sum_{n \geqslant 0} z^n = (1-z)^{-1}</syntaxhighlight> | ||
| [[Generalized hypergeometric function#math.70.58| Generalized hypergeometric function]] | | [[Generalized hypergeometric function#math.70.58| Generalized hypergeometric function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 1,983: | Line 1,985: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 21 | | [[Gold 21]] | ||
| <syntaxhighlight lang="tex" inline >\chi(-1) = 1</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\chi(-1) = 1</syntaxhighlight> | ||
| [[Dirichlet L-function#math.71.1| Dirichlet L-function]] | | [[Dirichlet L-function#math.71.1| Dirichlet L-function]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 2,063: | Line 2,065: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 22 | | [[Gold 22]] | ||
| <syntaxhighlight lang="tex" inline >\operatorname{Bi}'(z)\sim \frac{z^{\frac{1}{4}}e^{\frac{2}{3}z^{\frac{3}{2}}}}{\sqrt\pi\,}\left[ \sum_{n=0}^{\infty}\frac{1+6n}{1-6n} \dfrac{ \Gamma(n+\frac{5}{6})\Gamma(n+\frac{1}{6})\left(\frac{3}{4}\right)^n}{2\pi n! z^{3n/2}} \right]</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\operatorname{Bi}'(z)\sim \frac{z^{\frac{1}{4}}e^{\frac{2}{3}z^{\frac{3}{2}}}}{\sqrt\pi\,}\left[ \sum_{n=0}^{\infty}\frac{1+6n}{1-6n} \dfrac{ \Gamma(n+\frac{5}{6})\Gamma(n+\frac{1}{6})\left(\frac{3}{4}\right)^n}{2\pi n! z^{3n/2}} \right]</syntaxhighlight> | ||
| [[Airy function#math.72.15| Airy function]] | | [[Airy function#math.72.15| Airy function]] | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| Line 2,143: | Line 2,145: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 23 | | [[Gold 23]] | ||
| <syntaxhighlight lang="tex" inline >F'(y)=1-2yF(y)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >F'(y)=1-2yF(y)</syntaxhighlight> | ||
| [[Dawson function#math.73.41| Dawson function]] | | [[Dawson function#math.73.41| Dawson function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 2,200: | Line 2,202: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 24 | | [[Gold 24]] | ||
| <syntaxhighlight lang="tex" inline >s\not =1</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >s\not =1</syntaxhighlight> | ||
| [[Hurwitz zeta function#math.74.0| Hurwitz zeta function]] | | [[Hurwitz zeta function#math.74.0| Hurwitz zeta function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,252: | Line 2,254: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 25 | | [[Gold 25]] | ||
| <syntaxhighlight lang="tex" inline >q = e^{i\pi\tau}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >q = e^{i\pi\tau}</syntaxhighlight> | ||
| [[Theta function#math.75.6| Theta function]] | | [[Theta function#math.75.6| Theta function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,317: | Line 2,319: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 26 | | [[Gold 26]] | ||
| <syntaxhighlight lang="tex" inline >\frac{\mathrm{d}}{\mathrm{d}z} \operatorname{dn}(z) = - k^2 \operatorname{sn}(z) \operatorname{cn}(z)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\frac{\mathrm{d}}{\mathrm{d}z} \operatorname{dn}(z) = - k^2 \operatorname{sn}(z) \operatorname{cn}(z)</syntaxhighlight> | ||
| [[Jacobi elliptic functions#math.76.155| Jacobi elliptic functions]] | | [[Jacobi elliptic functions#math.76.155| Jacobi elliptic functions]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,391: | Line 2,393: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 27 | | [[Gold 27]] | ||
| <syntaxhighlight lang="tex" inline >\int_{-\infty}^\infty \frac {\gamma\left(\frac s 2, z^2 \pi \right)} {(z^2 \pi)^\frac s 2} e^{-2 \pi i k z} \mathrm d z = \frac {\Gamma\left(\frac {1-s} 2, k^2 \pi \right)} {(k^2 \pi)^\frac {1-s} 2}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\int_{-\infty}^\infty \frac {\gamma\left(\frac s 2, z^2 \pi \right)} {(z^2 \pi)^\frac s 2} e^{-2 \pi i k z} \mathrm d z = \frac {\Gamma\left(\frac {1-s} 2, k^2 \pi \right)} {(k^2 \pi)^\frac {1-s} 2}</syntaxhighlight> | ||
| [[Incomplete gamma function#math.77.118| Incomplete gamma function]] | | [[Incomplete gamma function#math.77.118| Incomplete gamma function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,466: | Line 2,468: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 28 | | [[Gold 28]] | ||
| <syntaxhighlight lang="tex" inline >_{1}(z) =</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >_{1}(z) =</syntaxhighlight> | ||
| [[Polylogarithm#math.78.0| Polylogarithm]] | | [[Polylogarithm#math.78.0| Polylogarithm]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 2,542: | Line 2,544: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 29 | | [[Gold 29]] | ||
| <syntaxhighlight lang="tex" inline >\int_{-\infty}^\infty \operatorname{sinc}(t) \, e^{-i 2 \pi f t}\,dt = \operatorname{rect}(f)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\int_{-\infty}^\infty \operatorname{sinc}(t) \, e^{-i 2 \pi f t}\,dt = \operatorname{rect}(f)</syntaxhighlight> | ||
| [[Sinc function#math.79.11| Sinc function]] | | [[Sinc function#math.79.11| Sinc function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,613: | Line 2,615: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 30 | | [[Gold 30]] | ||
| <syntaxhighlight lang="tex" inline >N=1</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >N=1</syntaxhighlight> | ||
| [[Exponential integral#math.80.26| Exponential integral]] | | [[Exponential integral#math.80.26| Exponential integral]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,670: | Line 2,672: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 31 | | [[Gold 31]] | ||
| <syntaxhighlight lang="tex" inline >\sum_{n=0}^\infty \frac{n!\,\Gamma\left(\alpha + 1\right)}{\Gamma\left(n+\alpha+1\right)}L_n^{(\alpha)}(x)L_n^{(\alpha)}(y)t^n=\frac{1}{(1-t)^{\alpha + 1}}e^{-(x+y)t/(1-t)}\,_0F_1\left(;\alpha + 1;\frac{xyt}{(1-t)^2}\right)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\sum_{n=0}^\infty \frac{n!\,\Gamma\left(\alpha + 1\right)}{\Gamma\left(n+\alpha+1\right)}L_n^{(\alpha)}(x)L_n^{(\alpha)}(y)t^n=\frac{1}{(1-t)^{\alpha + 1}}e^{-(x+y)t/(1-t)}\,_0F_1\left(;\alpha + 1;\frac{xyt}{(1-t)^2}\right)</syntaxhighlight> | ||
| [[Laguerre polynomials#math.81.84| Laguerre polynomials]] | | [[Laguerre polynomials#math.81.84| Laguerre polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 2,742: | Line 2,744: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 32 | | [[Gold 32]] | ||
| <syntaxhighlight lang="tex" inline >c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!}</syntaxhighlight> | ||
| [[Associated Legendre polynomials#math.82.8| Associated Legendre polynomials]] | | [[Associated Legendre polynomials#math.82.8| Associated Legendre polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,797: | Line 2,799: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 33 | | [[Gold 33]] | ||
| <syntaxhighlight lang="tex" inline >\mathrm{Gi}(x) = \frac{1}{\pi} \int_0^\infty \sin\left(\frac{t^3}{3} + xt\right)\, dt</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\mathrm{Gi}(x) = \frac{1}{\pi} \int_0^\infty \sin\left(\frac{t^3}{3} + xt\right)\, dt</syntaxhighlight> | ||
| [[Scorer's function#math.83.3| Scorer's function]] | | [[Scorer's function#math.83.3| Scorer's function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 2,883: | Line 2,885: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 34 | | [[Gold 34]] | ||
| <syntaxhighlight lang="tex" inline >\frac{\partial^2}{\partial x^2} V(x;\sigma,\gamma)= \frac{x^2-\gamma^2-\sigma^2}{\sigma^4} \frac{\operatorname{Re}[w(z)]}{\sigma\sqrt{2 \pi}}-\frac{2 x \gamma}{\sigma^4} \frac{\operatorname{Im}[w(z)]}{\sigma\sqrt{2 \pi}}+\frac{\gamma}{\sigma^4}\frac{1}{\pi}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\frac{\partial^2}{\partial x^2} V(x;\sigma,\gamma)= \frac{x^2-\gamma^2-\sigma^2}{\sigma^4} \frac{\operatorname{Re}[w(z)]}{\sigma\sqrt{2 \pi}}-\frac{2 x \gamma}{\sigma^4} \frac{\operatorname{Im}[w(z)]}{\sigma\sqrt{2 \pi}}+\frac{\gamma}{\sigma^4}\frac{1}{\pi}</syntaxhighlight> | ||
| [[Voigt profile#math.84.31| Voigt profile]] | | [[Voigt profile#math.84.31| Voigt profile]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 2,957: | Line 2,959: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 35 | | [[Gold 35]] | ||
| <syntaxhighlight lang="tex" inline >\Phi(z,s,a) = \frac{1}{1-z} \frac{1}{a^{s}} + \sum_{n=1}^{N-1} \frac{(-1)^{n} \mathrm{Li}_{-n}(z)}{n!} \frac{(s)_{n}}{a^{n+s}} +O(a^{-N-s})</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\Phi(z,s,a) = \frac{1}{1-z} \frac{1}{a^{s}} + \sum_{n=1}^{N-1} \frac{(-1)^{n} \mathrm{Li}_{-n}(z)}{n!} \frac{(s)_{n}}{a^{n+s}} +O(a^{-N-s})</syntaxhighlight> | ||
| [[Lerch zeta function#math.85.57| Lerch zeta function]] | | [[Lerch zeta function#math.85.57| Lerch zeta function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 3,030: | Line 3,032: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 36 | | [[Gold 36]] | ||
| <syntaxhighlight lang="tex" inline >M(1,2,z)=(e^z-1)/z,\ \ M(1,3,z)=2!(e^z-1-z)/z^2</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >M(1,2,z)=(e^z-1)/z,\ \ M(1,3,z)=2!(e^z-1-z)/z^2</syntaxhighlight> | ||
| [[Confluent hypergeometric function#math.86.44| Confluent hypergeometric function]] | | [[Confluent hypergeometric function#math.86.44| Confluent hypergeometric function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,101: | Line 3,103: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 37 | | [[Gold 37]] | ||
| <syntaxhighlight lang="tex" inline >\sigma = \pm 1</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\sigma = \pm 1</syntaxhighlight> | ||
| [[Mathieu function#math.87.54| Mathieu function]] | | [[Mathieu function#math.87.54| Mathieu function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,178: | Line 3,180: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 38 | | [[Gold 38]] | ||
| <syntaxhighlight lang="tex" inline >\frac{d^2f}{dz^2} + \left(\tilde{a}z^2+\tilde{b}z+\tilde{c}\right)f=0</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\frac{d^2f}{dz^2} + \left(\tilde{a}z^2+\tilde{b}z+\tilde{c}\right)f=0</syntaxhighlight> | ||
| [[Parabolic cylinder function#math.88.0| Parabolic cylinder function]] | | [[Parabolic cylinder function#math.88.0| Parabolic cylinder function]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 3,240: | Line 3,242: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 39 | | [[Gold 39]] | ||
| <syntaxhighlight lang="tex" inline >c=\infty</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >c=\infty</syntaxhighlight> | ||
| [[Painlevé transcendents#math.89.23| Painlevé transcendents]] | | [[Painlevé transcendents#math.89.23| Painlevé transcendents]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,305: | Line 3,307: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 40 | | [[Gold 40]] | ||
| <syntaxhighlight lang="tex" inline >c = a + 1</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >c = a + 1</syntaxhighlight> | ||
| [[Hypergeometric function#math.90.7| Hypergeometric function]] | | [[Hypergeometric function#math.90.7| Hypergeometric function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,359: | Line 3,361: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 41 | | [[Gold 41]] | ||
| <syntaxhighlight lang="tex" inline >\frac{1}{\Gamma(z)}= z e^{\gamma z} \prod_{k=1}^\infty \left\{ \left(1+\frac{z}{k}\right)e^{-z/k} \right\}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\frac{1}{\Gamma(z)}= z e^{\gamma z} \prod_{k=1}^\infty \left\{ \left(1+\frac{z}{k}\right)e^{-z/k} \right\}</syntaxhighlight> | ||
| [[Barnes G-function#math.91.47| Barnes G-function]] | | [[Barnes G-function#math.91.47| Barnes G-function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,451: | Line 3,453: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 42 | | [[Gold 42]] | ||
| <syntaxhighlight lang="tex" inline >192/24 = 8 = 2 \times 4</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >192/24 = 8 = 2 \times 4</syntaxhighlight> | ||
| [[Heun function#math.92.1| Heun function]] | | [[Heun function#math.92.1| Heun function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,691: | Line 3,693: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 43 | | [[Gold 43]] | ||
| <syntaxhighlight lang="tex" inline >=2</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >=2</syntaxhighlight> | ||
| [[Gegenbauer polynomials#math.93.0| Gegenbauer polynomials]] | | [[Gegenbauer polynomials#math.93.0| Gegenbauer polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 3,744: | Line 3,746: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 44 | | [[Gold 44]] | ||
| <syntaxhighlight lang="tex" inline >\lim_{q\to 1}\;_{j}\phi_k \left[\begin{matrix} q^{a_1} & q^{a_2} & \ldots & q^{a_j} \\ q^{b_1} & q^{b_2} & \ldots & q^{b_k} \end{matrix} ; q,(q-1)^{1+k-j} z \right]=\;_{j}F_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_j \\ b_1 & b_2 & \ldots & b_k \end{matrix} ;z \right]</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\lim_{q\to 1}\;_{j}\phi_k \left[\begin{matrix} q^{a_1} & q^{a_2} & \ldots & q^{a_j} \\ q^{b_1} & q^{b_2} & \ldots & q^{b_k} \end{matrix} ; q,(q-1)^{1+k-j} z \right]=\;_{j}F_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_j \\ b_1 & b_2 & \ldots & b_k \end{matrix} ;z \right]</syntaxhighlight> | ||
| [[Basic hypergeometric series#math.94.4| Basic hypergeometric series]] | | [[Basic hypergeometric series#math.94.4| Basic hypergeometric series]] | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| Line 3,802: | Line 3,804: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 45 | | [[Gold 45]] | ||
| <syntaxhighlight lang="tex" inline >\frac{d^2w}{dz^2}+\left(-\frac{1}{4}+\frac{\kappa}{z}+\frac{1/4-\mu^2}{z^2}\right)w=0</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\frac{d^2w}{dz^2}+\left(-\frac{1}{4}+\frac{\kappa}{z}+\frac{1/4-\mu^2}{z^2}\right)w=0</syntaxhighlight> | ||
| [[Whittaker function#math.95.0| Whittaker function]] | | [[Whittaker function#math.95.0| Whittaker function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,861: | Line 3,863: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 46 | | [[Gold 46]] | ||
| <syntaxhighlight lang="tex" inline >e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12</syntaxhighlight> | ||
| [[Lemniscatic elliptic function#math.96.1| Lemniscatic elliptic function]] | | [[Lemniscatic elliptic function#math.96.1| Lemniscatic elliptic function]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 3,916: | Line 3,918: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 47 | | [[Gold 47]] | ||
| <syntaxhighlight lang="tex" inline >\gamma> 0,n-p=m-q> 0</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\gamma> 0,n-p=m-q> 0</syntaxhighlight> | ||
| [[Meijer G-function#math.98.53| Meijer G-function]] | | [[Meijer G-function#math.98.53| Meijer G-function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 3,974: | Line 3,976: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 48 | | [[Gold 48]] | ||
| <syntaxhighlight lang="tex" inline >\begin{pmatrix} j \\ m \quad m'\end{pmatrix}:= \sqrt{2 j + 1}\begin{pmatrix} j & 0 & j \\ m & 0 & m'\end{pmatrix}= (-1)^{j - m'} \delta_{m, -m'}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\begin{pmatrix} j \\ m \quad m'\end{pmatrix}:= \sqrt{2 j + 1}\begin{pmatrix} j & 0 & j \\ m & 0 & m'\end{pmatrix}= (-1)^{j - m'} \delta_{m, -m'}</syntaxhighlight> | ||
| [[3-j symbol#math.99.30| 3-j symbol]] | | [[3-j symbol#math.99.30| 3-j symbol]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 4,033: | Line 4,035: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 49 | | [[Gold 49]] | ||
| <syntaxhighlight lang="tex" inline >\begin{Bmatrix} i & j & \ell\\ k & m & n \end{Bmatrix}= (\Phi_{i,j}^{k,m})_{\ell,n}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\begin{Bmatrix} i & j & \ell\\ k & m & n \end{Bmatrix}= (\Phi_{i,j}^{k,m})_{\ell,n}</syntaxhighlight> | ||
| [[6-j symbol#math.100.14| 6-j symbol]] | | [[6-j symbol#math.100.14| 6-j symbol]] | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| Line 4,099: | Line 4,101: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 50 | | [[Gold 50]] | ||
| <syntaxhighlight lang="tex" inline >\sum_{j_7 j_8} (2j_7+1)(2j_8+1) \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} \begin{Bmatrix} j_1 & j_2 & j_3'\\ j_4 & j_5 & j_6'\\ j_7 & j_8 & j_9 \end{Bmatrix} = \frac{\delta_{j_3j_3'}\delta_{j_6j_6'} \begin{Bmatrix} j_1 & j_2 & j_3 \end{Bmatrix} \begin{Bmatrix} j_4 & j_5 & j_6\end{Bmatrix} \begin{Bmatrix} j_3 & j_6 & j_9 \end{Bmatrix}} {(2j_3+1)(2j_6+1)}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\sum_{j_7 j_8} (2j_7+1)(2j_8+1) \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} \begin{Bmatrix} j_1 & j_2 & j_3'\\ j_4 & j_5 & j_6'\\ j_7 & j_8 & j_9 \end{Bmatrix} = \frac{\delta_{j_3j_3'}\delta_{j_6j_6'} \begin{Bmatrix} j_1 & j_2 & j_3 \end{Bmatrix} \begin{Bmatrix} j_4 & j_5 & j_6\end{Bmatrix} \begin{Bmatrix} j_3 & j_6 & j_9 \end{Bmatrix}} {(2j_3+1)(2j_6+1)}</syntaxhighlight> | ||
| [[9-j symbol#math.101.32| 9-j symbol]] | | [[9-j symbol#math.101.32| 9-j symbol]] | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| Line 4,158: | Line 4,160: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 51 | | [[Gold 51]] | ||
| <syntaxhighlight lang="tex" inline >\mathcal{K}_k(x; n,q) = \sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \binom {n-j}{k-j} \binom{x}{j}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\mathcal{K}_k(x; n,q) = \sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \binom {n-j}{k-j} \binom{x}{j}</syntaxhighlight> | ||
| [[Kravchuk polynomials#math.102.5| Kravchuk polynomials]] | | [[Kravchuk polynomials#math.102.5| Kravchuk polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 4,213: | Line 4,215: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 52 | | [[Gold 52]] | ||
| <syntaxhighlight lang="tex" inline >g_1(x) = \sum_{k \geq 1} \frac{\sin(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >g_1(x) = \sum_{k \geq 1} \frac{\sin(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2</syntaxhighlight> | ||
| [[Kelvin functions#math.103.8| Kelvin functions]] | | [[Kelvin functions#math.103.8| Kelvin functions]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 4,280: | Line 4,282: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 53 | | [[Gold 53]] | ||
| <syntaxhighlight lang="tex" inline >S_{\mu,\nu}(z) = s_{\mu,\nu}(z) + 2^{\mu-1} \Gamma\left(\frac{\mu + \nu + 1}{2}\right) \Gamma\left(\frac{\mu - \nu + 1}{2}\right)\left(\sin \left[(\mu - \nu)\frac{\pi}{2}\right] J_\nu(z) - \cos \left[(\mu - \nu)\frac{\pi}{2}\right] Y_\nu(z)\right)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >S_{\mu,\nu}(z) = s_{\mu,\nu}(z) + 2^{\mu-1} \Gamma\left(\frac{\mu + \nu + 1}{2}\right) \Gamma\left(\frac{\mu - \nu + 1}{2}\right)\left(\sin \left[(\mu - \nu)\frac{\pi}{2}\right] J_\nu(z) - \cos \left[(\mu - \nu)\frac{\pi}{2}\right] Y_\nu(z)\right)</syntaxhighlight> | ||
| [[Lommel function#math.104.2| Lommel function]] | | [[Lommel function#math.104.2| Lommel function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 4,344: | Line 4,346: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 54 | | [[Gold 54]] | ||
| <syntaxhighlight lang="tex" inline >\mathbf{H}_{\alpha}(z) = \frac{z^{\alpha+1}}{2^{\alpha}\sqrt{\pi} \Gamma \left (\alpha+\tfrac{3}{2} \right )} {}_1F_2 \left (1,\tfrac{3}{2}, \alpha+\tfrac{3}{2},-\tfrac{z^2}{4} \right )</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\mathbf{H}_{\alpha}(z) = \frac{z^{\alpha+1}}{2^{\alpha}\sqrt{\pi} \Gamma \left (\alpha+\tfrac{3}{2} \right )} {}_1F_2 \left (1,\tfrac{3}{2}, \alpha+\tfrac{3}{2},-\tfrac{z^2}{4} \right )</syntaxhighlight> | ||
| [[Struve function#math.105.18| Struve function]] | | [[Struve function#math.105.18| Struve function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 4,417: | Line 4,419: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 55 | | [[Gold 55]] | ||
| <syntaxhighlight lang="tex" inline >f(t+p) = f(t)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >f(t+p) = f(t)</syntaxhighlight> | ||
| [[Hill differential equation#math.106.7| Hill differential equation]] | | [[Hill differential equation#math.106.7| Hill differential equation]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 4,479: | Line 4,481: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 56 | | [[Gold 56]] | ||
| <syntaxhighlight lang="tex" inline >\mathbf{J}_\nu(z)=\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}+\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\mathbf{J}_\nu(z)=\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}+\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}</syntaxhighlight> | ||
| [[Anger function#math.108.3| Anger function]] | | [[Anger function#math.108.3| Anger function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 4,542: | Line 4,544: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 57 | | [[Gold 57]] | ||
| <syntaxhighlight lang="tex" inline >(\operatorname{Ec})^'_{2K} = (\operatorname{Ec})^'_0 = 0, \;\; (\operatorname{Es})^'_{2K} = (\operatorname{Es})^'_0 = 0</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >(\operatorname{Ec})^'_{2K} = (\operatorname{Ec})^'_0 = 0, \;\; (\operatorname{Es})^'_{2K} = (\operatorname{Es})^'_0 = 0</syntaxhighlight> | ||
| [[Lamé function#math.109.27| Lamé function]] | | [[Lamé function#math.109.27| Lamé function]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
| Line 4,593: | Line 4,595: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 58 | | [[Gold 58]] | ||
| <syntaxhighlight lang="tex" inline >\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)</syntaxhighlight> | ||
| [[Gauss–Hermite quadrature#math.110.1| Gauss–Hermite quadrature]] | | [[Gauss–Hermite quadrature#math.110.1| Gauss–Hermite quadrature]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
| Line 4,658: | Line 4,660: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 59 | | [[Gold 59]] | ||
| <syntaxhighlight lang="tex" inline >p_n(x;a,b,c,d|q) =(ab,ac,ad;q)_na^{-n}\;_{4}\phi_3 \left[\begin{matrix} q^{-n}&abcdq^{n-1}&ae^{i\theta}&ae^{-i\theta} \\ ab&ac&ad \end{matrix} ; q,q \right]</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >p_n(x;a,b,c,d|q) =(ab,ac,ad;q)_na^{-n}\;_{4}\phi_3 \left[\begin{matrix} q^{-n}&abcdq^{n-1}&ae^{i\theta}&ae^{-i\theta} \\ ab&ac&ad \end{matrix} ; q,q \right]</syntaxhighlight> | ||
| [[Askey–Wilson polynomials#math.111.0| Askey–Wilson polynomials]] | | [[Askey–Wilson polynomials#math.111.0| Askey–Wilson polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 4,718: | Line 4,720: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 60 | | [[Gold 60]] | ||
| <syntaxhighlight lang="tex" inline >Q_n(x;\alpha,\beta,N)= {}_3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1).</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >Q_n(x;\alpha,\beta,N)= {}_3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1).</syntaxhighlight> | ||
| [[Hahn polynomials#math.112.0| Hahn polynomials]] | | [[Hahn polynomials#math.112.0| Hahn polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 4,777: | Line 4,779: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 61 | | [[Gold 61]] | ||
| <syntaxhighlight lang="tex" inline >\sum_{x=0}^\infty \frac{\mu^x}{x!} C_n(x; \mu)C_m(x; \mu)=\mu^{-n} e^\mu n! \delta_{nm}, \quad \mu>0</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\sum_{x=0}^\infty \frac{\mu^x}{x!} C_n(x; \mu)C_m(x; \mu)=\mu^{-n} e^\mu n! \delta_{nm}, \quad \mu>0</syntaxhighlight> | ||
| [[Charlier polynomials#math.113.2| Charlier polynomials]] | | [[Charlier polynomials#math.113.2| Charlier polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 4,829: | Line 4,831: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 62 | | [[Gold 62]] | ||
| <syntaxhighlight lang="tex" inline >p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\phi_3\left[\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\\aq&bdq&cq\\ \end{matrix};q;q\right]</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\phi_3\left[\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\\aq&bdq&cq\\ \end{matrix};q;q\right]</syntaxhighlight> | ||
| [[Q-Racah polynomials#math.114.0| Q-Racah polynomials]] | | [[Q-Racah polynomials#math.114.0| Q-Racah polynomials]] | ||
| {{na}} | | {{na}} | ||
| - | |||
| - | | - | ||
| - | | - | ||
| Line 4,840: | Line 4,843: | ||
| - | | - | ||
| {{cross|20|color=orange}} | | {{cross|20|color=orange}} | ||
| Did not find q-Recah polynomial. Since it is not a definition, and q-Recah are not supported by Mathematica, there is no translation possible. | | Did not find q-Recah polynomial. Since it is not a definition, and q-Recah are not supported by Mathematica, there is no translation possible. | ||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | | <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | ||
| Line 4,873: | Line 4,875: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 63 | | [[Gold 63]] | ||
| <syntaxhighlight lang="tex" inline >\displaystyle c_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\displaystyle c_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a)</syntaxhighlight> | ||
| [[Q-Charlier polynomials#math.115.0| Q-Charlier polynomials]] | | [[Q-Charlier polynomials#math.115.0| Q-Charlier polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
| Line 4,885: | Line 4,887: | ||
| - | | - | ||
| - | | - | ||
| | | {{cross|20|color=orange}} | ||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | | <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | ||
"id": 63, | "id": 63, | ||
| Line 4,921: | Line 4,923: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 64 | | [[Gold 64]] | ||
| <syntaxhighlight lang="tex" inline >M_n(x,\beta,\gamma) = \sum_{k=0}^n (-1)^k{n \choose k}{x\choose k}k!(x+\beta)_{n-k}\gamma^{-k}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >M_n(x,\beta,\gamma) = \sum_{k=0}^n (-1)^k{n \choose k}{x\choose k}k!(x+\beta)_{n-k}\gamma^{-k}</syntaxhighlight> | ||
| [[Meixner polynomials#math.116.0| Meixner polynomials]] | | [[Meixner polynomials#math.116.0| Meixner polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 4,977: | Line 4,979: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 65 | | [[Gold 65]] | ||
| <syntaxhighlight lang="tex" inline >x(1-x) \frac {\partial^2F_1(x,y)} {\partial x^2} + y(1-x) \frac {\partial^2F_1(x,y)} {\partial x \partial y} + [c - (a+b_1+1) x] \frac {\partial F_1(x,y)} {\partial x} - b_1 y \frac {\partial F_1(x,y)} {\partial y} - a b_1 F_1(x,y) = 0</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >x(1-x) \frac {\partial^2F_1(x,y)} {\partial x^2} + y(1-x) \frac {\partial^2F_1(x,y)} {\partial x \partial y} + [c - (a+b_1+1) x] \frac {\partial F_1(x,y)} {\partial x} - b_1 y \frac {\partial F_1(x,y)} {\partial y} - a b_1 F_1(x,y) = 0</syntaxhighlight> | ||
| [[Appell series#math.117.19| Appell series]] | | [[Appell series#math.117.19| Appell series]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,047: | Line 5,049: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 66 | | [[Gold 66]] | ||
| <syntaxhighlight lang="tex" inline >\Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau > 0</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau > 0</syntaxhighlight> | ||
| [[Theta function of a lattice#math.118.0| Theta function of a lattice]] | | [[Theta function of a lattice#math.118.0| Theta function of a lattice]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,105: | Line 5,107: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 67 | | [[Gold 67]] | ||
| <syntaxhighlight lang="tex" inline >\frac{d^2 S}{dz^2}+\left(\sum _{j=1}^N \frac{\gamma _j}{z - a_j} \right) \frac{dS}{dz} + \frac{V(z)}{\prod _{j=1}^N (z - a_j)}S = 0</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\frac{d^2 S}{dz^2}+\left(\sum _{j=1}^N \frac{\gamma _j}{z - a_j} \right) \frac{dS}{dz} + \frac{V(z)}{\prod _{j=1}^N (z - a_j)}S = 0</syntaxhighlight> | ||
| [[Heine–Stieltjes polynomials#math.119.0| Heine–Stieltjes polynomials]] | | [[Heine–Stieltjes polynomials#math.119.0| Heine–Stieltjes polynomials]] | ||
| {{na}} | | {{na}} | ||
| - | |||
| - | | - | ||
| {{cross|20|color=orange}} | | {{cross|20|color=orange}} | ||
| Line 5,117: | Line 5,119: | ||
| - | | - | ||
| - | | - | ||
| Mistakenly detected Stieltjes constant. | | Mistakenly detected Stieltjes constant. No translation possible for S. | ||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | | <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | ||
"id": 67, | "id": 67, | ||
| Line 5,185: | Line 5,187: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 68 | | [[Gold 68]] | ||
| <syntaxhighlight lang="tex" inline >w(x) = \frac{k}{\sqrt{\pi}} x^{-1/2} \exp(-k^2\log^2 x)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >w(x) = \frac{k}{\sqrt{\pi}} x^{-1/2} \exp(-k^2\log^2 x)</syntaxhighlight> | ||
| [[Stieltjes–Wigert polynomials#math.120.0| Stieltjes–Wigert polynomials]] | | [[Stieltjes–Wigert polynomials#math.120.0| Stieltjes–Wigert polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{ | | {{na}} | ||
| | | {{cross|20|color=orange}} | ||
| - | | - | ||
| - | | - | ||
| Line 5,270: | Line 5,272: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 69 | | [[Gold 69]] | ||
| <syntaxhighlight lang="tex" inline >y^2=x(x-1)(x-\lambda)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >y^2=x(x-1)(x-\lambda)</syntaxhighlight> | ||
| [[Modular lambda function#math.121.23| Modular lambda function]] | | [[Modular lambda function#math.121.23| Modular lambda function]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 5,331: | Line 5,333: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 70 | | [[Gold 70]] | ||
| <syntaxhighlight lang="tex" inline >P_1^{(\lambda)}(x;\phi)=2(\lambda\cos\phi + x\sin\phi)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >P_1^{(\lambda)}(x;\phi)=2(\lambda\cos\phi + x\sin\phi)</syntaxhighlight> | ||
| [[Meixner–Pollaczek polynomials#math.122.3| Meixner–Pollaczek polynomials]] | | [[Meixner–Pollaczek polynomials#math.122.3| Meixner–Pollaczek polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
| Line 5,343: | Line 5,344: | ||
| - | | - | ||
| - | | - | ||
| {{cross|20|color=orange}} | |||
| - | | - | ||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | | <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | ||
| Line 5,377: | Line 5,379: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 71 | | [[Gold 71]] | ||
| <syntaxhighlight lang="tex" inline >P_n^{(\alpha,\beta)}(z)=\frac{(\alpha+1)_n}{n!}\,{}_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\tfrac{1}{2}(1-z)\right)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >P_n^{(\alpha,\beta)}(z)=\frac{(\alpha+1)_n}{n!}\,{}_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\tfrac{1}{2}(1-z)\right)</syntaxhighlight> | ||
| [[Jacobi polynomials#math.123.0| Jacobi polynomials]] | | [[Jacobi polynomials#math.123.0| Jacobi polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{ya}} | | {{ya}} | ||
| Line 5,444: | Line 5,446: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 72 | | [[Gold 72]] | ||
| <syntaxhighlight lang="tex" inline >S_n(x^2;a,b,c)= {}_3F_2(-n,a+ix,a-ix;a+b,a+c;1).</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >S_n(x^2;a,b,c)= {}_3F_2(-n,a+ix,a-ix;a+b,a+c;1).</syntaxhighlight> | ||
| [[Continuous dual Hahn polynomials#math.124.0| Continuous dual Hahn polynomials]] | | [[Continuous dual Hahn polynomials#math.124.0| Continuous dual Hahn polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
| Line 5,502: | Line 5,504: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 73 | | [[Gold 73]] | ||
| <syntaxhighlight lang="tex" inline >P_n^{(\alpha,\beta)}=\lim_{t\to\infty}t^{-n}p_n\left(\tfrac12xt; \tfrac12(\alpha+1-it), \tfrac12(\beta+1+it), \tfrac12(\alpha+1+it), \tfrac12(\beta+1-it)\right)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >P_n^{(\alpha,\beta)}=\lim_{t\to\infty}t^{-n}p_n\left(\tfrac12xt; \tfrac12(\alpha+1-it), \tfrac12(\beta+1+it), \tfrac12(\alpha+1+it), \tfrac12(\beta+1-it)\right)</syntaxhighlight> | ||
| [[Continuous Hahn polynomials#math.125.15| Continuous Hahn polynomials]] | | [[Continuous Hahn polynomials#math.125.15| Continuous Hahn polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,562: | Line 5,564: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 74 | | [[Gold 74]] | ||
| <syntaxhighlight lang="tex" inline >\sum^{b-1}_{s=a}w_n^{(c)}(s,a,b)w_m^{(c)}(s,a,b)\rho(s)[\Delta x(s-\frac{1}{2}) ]=\delta_{nm}d_n^2</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\sum^{b-1}_{s=a}w_n^{(c)}(s,a,b)w_m^{(c)}(s,a,b)\rho(s)[\Delta x(s-\frac{1}{2}) ]=\delta_{nm}d_n^2</syntaxhighlight> | ||
| [[Dual Hahn polynomials#math.126.7| Dual Hahn polynomials]] | | [[Dual Hahn polynomials#math.126.7| Dual Hahn polynomials]] | ||
| {{na}} | | {{na}} | ||
| - | |||
| - | | - | ||
| {{cross|20|color=orange}} | | {{cross|20|color=orange}} | ||
| Line 5,574: | Line 5,576: | ||
| - | | - | ||
| - | | - | ||
| Not standard notation for dual Hahn polynomial. DLMF uses {{math|R}}. | | Not standard notation for dual Hahn polynomial. DLMF uses {{math|R}}. Further, dual Hahn does not exist in Mathematica. | ||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | | <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | ||
"id": 74, | "id": 74, | ||
| Line 5,608: | Line 5,610: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 75 | | [[Gold 75]] | ||
| <syntaxhighlight lang="tex" inline >p_n(x;a,b,c|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_n*_4\Phi_3(q^{-n},abcdq^{n-1},ae^{i{(t+2u)}},ae^{-it};abe^{2iu},ac,ad;q;q)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >p_n(x;a,b,c|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_n*_4\Phi_3(q^{-n},abcdq^{n-1},ae^{i{(t+2u)}},ae^{-it};abe^{2iu},ac,ad;q;q)</syntaxhighlight> | ||
| [[Continuous q-Hahn polynomials#math.127.0| Continuous q-Hahn polynomials]] | | [[Continuous q-Hahn polynomials#math.127.0| Continuous q-Hahn polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,674: | Line 5,676: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 76 | | [[Gold 76]] | ||
| <syntaxhighlight lang="tex" inline >p_n(x;a,b,c\mid q)=\frac{(ab,ac;q)_n}{a^n}\cdot {_3\Phi_2}(q^-n,ae^{i\theta},ae^{-i\theta}; ab, ac \mid q;q)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >p_n(x;a,b,c\mid q)=\frac{(ab,ac;q)_n}{a^n}\cdot {_3\Phi_2}(q^-n,ae^{i\theta},ae^{-i\theta}; ab, ac \mid q;q)</syntaxhighlight> | ||
| [[Continuous dual q-Hahn polynomials#math.128.0| Continuous dual q-Hahn polynomials]] | | [[Continuous dual q-Hahn polynomials#math.128.0| Continuous dual q-Hahn polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,685: | Line 5,687: | ||
| - | | - | ||
| - | | - | ||
| - | | Wrong LaTeX. <code>q^-n</code> only puts <math>-</math> into the subscript but not <math>-n</math>. | ||
| Underscore mismatch. | | Underscore mismatch. | ||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | | <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | ||
| Line 5,696: | Line 5,698: | ||
"id": "FORMULA_95daf919f18506606090e49a38d1c1a6", | "id": "FORMULA_95daf919f18506606090e49a38d1c1a6", | ||
"formula": "p_n(x;a,b,c\\mid q)=\\frac{(ab,ac;q)_n}{a^n}\\cdot {_3\\Phi_2}(q^-n,ae^{i\\theta},ae^{-i\\theta}; ab, ac \\mid q;q)", | "formula": "p_n(x;a,b,c\\mid q)=\\frac{(ab,ac;q)_n}{a^n}\\cdot {_3\\Phi_2}(q^-n,ae^{i\\theta},ae^{-i\\theta}; ab, ac \\mid q;q)", | ||
"semanticFormula": "p_n(x ; a , b , c | "semanticFormula": "p_n(x ; a , b , c \mid q) = \frac{\qmultiPochhammersym{ab , ac}{q}{n}}{a^n} \cdot \qgenhyperphi{3}{2}@{q^{- n} , a\expe^{\iunit \theta} , a\expe^{- \iunit \theta}}{ab , ac}{q}{q}", | ||
"confidence": 0.8662724998444776, | "confidence": 0.8662724998444776, | ||
"translations": { | "translations": { | ||
"Mathematica": { | "Mathematica": { | ||
"translation": "p[n_, x_, a_, b_, c_, q_] := Divide[Product[QPochhammer[Part[{a*b , a*c},i],q,n],{i,1,Length[{a*b , a*c}]}],(a)^(n)] * QHypergeometricPFQ[{(q)^(-) | "translation": "p[n_, x_, a_, b_, c_, q_] := Divide[Product[QPochhammer[Part[{a*b , a*c},i],q,n],{i,1,Length[{a*b , a*c}]}],(a)^(n)] * QHypergeometricPFQ[{(q)^(- n), a*Exp[I*\[Theta]], a*Exp[- I*\[Theta]]},{a*b , a*c},q,q]" | ||
} | } | ||
}, | }, | ||
| Line 5,736: | Line 5,738: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 77 | | [[Gold 77]] | ||
| <syntaxhighlight lang="tex" inline >Q_n(x;a,b,N;q)=\;_{3}\phi_2\left[\begin{matrix} q^-n & abq^n+1 & x \\ aq & q^-N \end{matrix} ; q,q \right]</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >Q_n(x;a,b,N;q)=\;_{3}\phi_2\left[\begin{matrix} q^-n & abq^n+1 & x \\ aq & q^-N \end{matrix} ; q,q \right]</syntaxhighlight> | ||
| [[Q-Hahn polynomials#math.129.0| Q-Hahn polynomials]] | | [[Q-Hahn polynomials#math.129.0| Q-Hahn polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,796: | Line 5,798: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 78 | | [[Gold 78]] | ||
| <syntaxhighlight lang="tex" inline >x=</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >x=</syntaxhighlight> | ||
| [[Al-Salam–Chihara polynomials#math.131.0| Al-Salam–Chihara polynomials]] | | [[Al-Salam–Chihara polynomials#math.131.0| Al-Salam–Chihara polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,851: | Line 5,853: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 79 | | [[Gold 79]] | ||
| <syntaxhighlight lang="tex" inline >\Phi_n^*(z)=z^n\overline{\Phi_n(1/\overline{z})}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\Phi_n^*(z)=z^n\overline{\Phi_n(1/\overline{z})}</syntaxhighlight> | ||
| [[Orthogonal polynomials on the unit circle#math.132.7| Orthogonal polynomials on the unit circle]] | | [[Orthogonal polynomials on the unit circle#math.132.7| Orthogonal polynomials on the unit circle]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 5,903: | Line 5,905: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 80 | | [[Gold 80]] | ||
| <syntaxhighlight lang="tex" inline >P_n(x) = c_n \, \det \begin{bmatrix}m_0 & m_1 & m_2 &\cdots & m_n \\m_1 & m_2 & m_3 &\cdots & m_{n+1} \\&&\vdots&& \vdots \\m_{n-1} &m_n& m_{n+1} &\cdots &m_{2n-1}\\1 & x & x^2 & \cdots & x^n\end{bmatrix}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >P_n(x) = c_n \, \det \begin{bmatrix}m_0 & m_1 & m_2 &\cdots & m_n \\m_1 & m_2 & m_3 &\cdots & m_{n+1} \\&&\vdots&& \vdots \\m_{n-1} &m_n& m_{n+1} &\cdots &m_{2n-1}\\1 & x & x^2 & \cdots & x^n\end{bmatrix}</syntaxhighlight> | ||
| [[Orthogonal polynomials#math.133.8| Orthogonal polynomials]] | | [[Orthogonal polynomials#math.133.8| Orthogonal polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
| Line 5,963: | Line 5,965: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 81 | | [[Gold 81]] | ||
| <syntaxhighlight lang="tex" inline >\displaystyle p_n(x;a,b;q) = {}_2\phi_1(q^{-n},abq^{n+1};aq;q,xq)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\displaystyle p_n(x;a,b;q) = {}_2\phi_1(q^{-n},abq^{n+1};aq;q,xq)</syntaxhighlight> | ||
| [[Little q-Jacobi polynomials#math.134.0| Little q-Jacobi polynomials]] | | [[Little q-Jacobi polynomials#math.134.0| Little q-Jacobi polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 6,026: | Line 6,028: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 82 | | [[Gold 82]] | ||
| <syntaxhighlight lang="tex" inline >\displaystyle P_n(x;a,b,c;q)={}_3\phi_2(q^{-n},abq^{n+1},x;aq,cq;q,q)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\displaystyle P_n(x;a,b,c;q)={}_3\phi_2(q^{-n},abq^{n+1},x;aq,cq;q,q)</syntaxhighlight> | ||
| [[Big q-Jacobi polynomials#math.135.0| Big q-Jacobi polynomials]] | | [[Big q-Jacobi polynomials#math.135.0| Big q-Jacobi polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 6,085: | Line 6,087: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 83 | | [[Gold 83]] | ||
| <syntaxhighlight lang="tex" inline >P_n(x;a,b;q)=\frac{1}{(b^{-1}*q^{-n};q,n)}*_2\Phi_1(q^{-n},aqx^{-1};aq|q;\frac{x}{b})</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >P_n(x;a,b;q)=\frac{1}{(b^{-1}*q^{-n};q,n)}*_2\Phi_1(q^{-n},aqx^{-1};aq|q;\frac{x}{b})</syntaxhighlight> | ||
| [[Big q-Laguerre polynomials#math.137.0| Big q-Laguerre polynomials]] | | [[Big q-Laguerre polynomials#math.137.0| Big q-Laguerre polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 6,151: | Line 6,153: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 84 | | [[Gold 84]] | ||
| <syntaxhighlight lang="tex" inline >K_n(\lambda(x);c,N|q)=_3\Phi_2(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >K_n(\lambda(x);c,N|q)=_3\Phi_2(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q)</syntaxhighlight> | ||
| [[Dual q-Krawtchouk polynomials#math.138.0| Dual q-Krawtchouk polynomials]] | | [[Dual q-Krawtchouk polynomials#math.138.0| Dual q-Krawtchouk polynomials]] | ||
| {{na}} | | {{na}} | ||
| - | |||
| - | | - | ||
| {{cross|20|color=orange}} | | {{cross|20|color=orange}} | ||
| Line 6,163: | Line 6,165: | ||
| - | | - | ||
| - | | - | ||
| Illegal LaTeX. Equal sign has underscore 3 (which is wrong). | | Illegal LaTeX. Equal sign has underscore 3 (which is wrong). Further, dual q-Krawtchouk do not exist in Mathematica. | ||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | | <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | ||
"id": 84, | "id": 84, | ||
| Line 6,209: | Line 6,211: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 85 | | [[Gold 85]] | ||
| <syntaxhighlight lang="tex" inline >P_{n}^{(\alpha)}(x|q)=\frac{(q^\alpha+1;q)_{n}}{(q;q)_{n}}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >P_{n}^{(\alpha)}(x|q)=\frac{(q^\alpha+1;q)_{n}}{(q;q)_{n}}</syntaxhighlight> | ||
| [[Continuous q-Laguerre polynomials#math.139.0| Continuous q-Laguerre polynomials]] | | [[Continuous q-Laguerre polynomials#math.139.0| Continuous q-Laguerre polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 6,267: | Line 6,269: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 86 | | [[Gold 86]] | ||
| <syntaxhighlight lang="tex" inline >\displaystyle p_n(x;a|q) = {}_2\phi_1(q^{-n},0;aq;q,qx) = \frac{1}{(a^{-1}q^{-n};q)_n}{}_2\phi_0(q^{-n},x^{-1};;q,x/a)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\displaystyle p_n(x;a|q) = {}_2\phi_1(q^{-n},0;aq;q,qx) = \frac{1}{(a^{-1}q^{-n};q)_n}{}_2\phi_0(q^{-n},x^{-1};;q,x/a)</syntaxhighlight> | ||
| [[Little q-Laguerre polynomials#math.142.0| Little q-Laguerre polynomials]] | | [[Little q-Laguerre polynomials#math.142.0| Little q-Laguerre polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 6,332: | Line 6,334: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 87 | | [[Gold 87]] | ||
| <syntaxhighlight lang="tex" inline >y_{n}(x;a;q)=\;_{2}\phi_1 \left(\begin{matrix} q^{-N} & -aq^{n} \\ 0 \end{matrix} ; q,qx \right)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >y_{n}(x;a;q)=\;_{2}\phi_1 \left(\begin{matrix} q^{-N} & -aq^{n} \\ 0 \end{matrix} ; q,qx \right)</syntaxhighlight> | ||
| [[Q-Bessel polynomials#math.143.0| Q-Bessel polynomials]] | | [[Q-Bessel polynomials#math.143.0| Q-Bessel polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 6,390: | Line 6,392: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 88 | | [[Gold 88]] | ||
| <syntaxhighlight lang="tex" inline >h_n(ix;q^{-1}) = i^n\hat h_n(x;q)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >h_n(ix;q^{-1}) = i^n\hat h_n(x;q)</syntaxhighlight> | ||
| [[Discrete q-Hermite polynomials#math.144.2| Discrete q-Hermite polynomials]] | | [[Discrete q-Hermite polynomials#math.144.2| Discrete q-Hermite polynomials]] | ||
| {{na}} | | {{na}} | ||
| - | |||
| - | | - | ||
| {{cross|20|color=orange}} | | {{cross|20|color=orange}} | ||
| Line 6,402: | Line 6,404: | ||
| - | | - | ||
| - | | - | ||
| We correctly identified <code>\discqHermitepolyhI</code> but were not able to distinguish it from <code>discqHermitepolyhII</code> from RHS. | | We correctly identified <code>\discqHermitepolyhI</code> but were not able to distinguish it from <code>discqHermitepolyhII</code> from RHS. Neither of them is translatable though. | ||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | | <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | ||
"id": 88, | "id": 88, | ||
| Line 6,454: | Line 6,456: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 89 | | [[Gold 89]] | ||
| <syntaxhighlight lang="tex" inline >P_{n}(x;a\mid q) = a^{-n} e^{in\phi} \frac{a^2;q_n}{(q;q)_n} {_3}\Phi_2(q^-n, ae^{i(\theta+2\phi)}, ae^{-i\theta}; a^2, 0 \mid q; q)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >P_{n}(x;a\mid q) = a^{-n} e^{in\phi} \frac{a^2;q_n}{(q;q)_n} {_3}\Phi_2(q^-n, ae^{i(\theta+2\phi)}, ae^{-i\theta}; a^2, 0 \mid q; q)</syntaxhighlight> | ||
| [[Q-Meixner–Pollaczek polynomials#math.145.0| Q-Meixner–Pollaczek polynomials]] | | [[Q-Meixner–Pollaczek polynomials#math.145.0| Q-Meixner–Pollaczek polynomials]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 6,518: | Line 6,520: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 90 | | [[Gold 90]] | ||
| <syntaxhighlight lang="tex" inline >\displaystyle L_n^{(\alpha)}(x;q) = \frac{(q^{\alpha+1};q)_n}{(q;q)_n} {}_1\phi_1(q^{-n};q^{\alpha+1};q,-q^{n+\alpha+1}x)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\displaystyle L_n^{(\alpha)}(x;q) = \frac{(q^{\alpha+1};q)_n}{(q;q)_n} {}_1\phi_1(q^{-n};q^{\alpha+1};q,-q^{n+\alpha+1}x)</syntaxhighlight> | ||
| [[Q-Laguerre polynomials#math.149.0| Q-Laguerre polynomials]] | | [[Q-Laguerre polynomials#math.149.0| Q-Laguerre polynomials]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 6,585: | Line 6,587: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 91 | | [[Gold 91]] | ||
| <syntaxhighlight lang="tex" inline >\sum_{n=0}^\infty H_n(x \mid q) \frac{t^n}{(q;q)_n} = \frac{1}{\left( t e^{i \theta},t e^{-i \theta};q \right)_\infty}</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >\sum_{n=0}^\infty H_n(x \mid q) \frac{t^n}{(q;q)_n} = \frac{1}{\left( t e^{i \theta},t e^{-i \theta};q \right)_\infty}</syntaxhighlight> | ||
| [[Continuous q-Hermite polynomials#math.150.3| Continuous q-Hermite polynomials]] | | [[Continuous q-Hermite polynomials#math.150.3| Continuous q-Hermite polynomials]] | ||
| {{na}} | | {{na}} | ||
| - | | - | ||
| Line 6,635: | Line 6,637: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 92 | | [[Gold 92]] | ||
| <syntaxhighlight lang="tex" inline >w^{\prime\prime}+\xi\sin(2z)w^{\prime}+(\eta-p\xi\cos(2z))w=0. </syntaxhighlight> | | <syntaxhighlight lang="tex" inline >w^{\prime\prime}+\xi\sin(2z)w^{\prime}+(\eta-p\xi\cos(2z))w=0. </syntaxhighlight> | ||
| [[Ince equation#math.151.0| Ince equation]] | | [[Ince equation#math.151.0| Ince equation]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
| Line 6,709: | Line 6,711: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 93 | | [[Gold 93]] | ||
| <syntaxhighlight lang="tex" inline >Q_v^\mu(x)= \cos(\mu\pi)\left(\frac{1+x}{1-x}\right)^{\mu/2}\frac{F(v+1,-v;1-\mu;1/2-2/x)} {\Gamma(1-\mu ) }</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >Q_v^\mu(x)= \cos(\mu\pi)\left(\frac{1+x}{1-x}\right)^{\mu/2}\frac{F(v+1,-v;1-\mu;1/2-2/x)} {\Gamma(1-\mu ) }</syntaxhighlight> | ||
| [[Ferrers function#math.152.1| Ferrers function]] | | [[Ferrers function#math.152.1| Ferrers function]] | ||
| {{na}} | | {{na}} | ||
| {{na}} | | {{na}} | ||
| Line 6,772: | Line 6,774: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 94 | | [[Gold 94]] | ||
| <syntaxhighlight lang="tex" inline >H_{-v}^{(1)}(z,w)=e^{v\pi i}H_v^{(1)}(z,w)</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >H_{-v}^{(1)}(z,w)=e^{v\pi i}H_v^{(1)}(z,w)</syntaxhighlight> | ||
| [[Incomplete Bessel functions#math.153.27| Incomplete Bessel functions]] | | [[Incomplete Bessel functions#math.153.27| Incomplete Bessel functions]] | ||
| {{ya}} | | {{ya}} | ||
| - | | - | ||
| Line 6,784: | Line 6,786: | ||
| - | | - | ||
| - | | - | ||
| | | {{cross|20|color=orange}} | ||
| <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | | <div class="toccolours mw-collapsible mw-collapsed" style="text-align: left">Full data: <div class="mw-collapsible-content"><syntaxhighlight lang="json" >{ | ||
"id": 94, | "id": 94, | ||
| Line 6,820: | Line 6,822: | ||
}</syntaxhighlight> | }</syntaxhighlight> | ||
</div></div> | </div></div> | ||
|- | |- | ||
| 95 | | [[Gold 95]] | ||
| <syntaxhighlight lang="tex" inline >K_v(x,y)=\int_1^\infty\frac{e^{-xt-\frac{y}{t}}}{t^{v+1}}dt</syntaxhighlight> | | <syntaxhighlight lang="tex" inline >K_v(x,y)=\int_1^\infty\frac{e^{-xt-\frac{y}{t}}}{t^{v+1}}dt</syntaxhighlight> | ||
| [[Incomplete Bessel K function/generalized incomplete gamma function#math.154.0| Incomplete Bessel K function/generalized incomplete gamma function]] | | [[Incomplete Bessel K function/generalized incomplete gamma function#math.154.0| Incomplete Bessel K function/generalized incomplete gamma function]] | ||
| {{ya}} | | {{ya}} | ||
| {{na}} | | {{na}} | ||
Latest revision as of 13:58, 1 September 2021
This is an overview table with the 95 benchmark entries showing the TeX of the formula and if the translation to semantic LaTeX and Mathematica (CAS) was correct. If the translation was wrong, a more detailed explanation and categorization of the error is given. This table does not contain the actual translation results. You can reach this information by clicking the link in the most left column, e.g., Gold 1.
We provide the entire benchmark as a JSON file too: gold-data.json.
| Entry Info | Translations | Reason For Failure | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| # | Formula | Title | Semantic LaTeX | CAS Translations | Definition / Substitution | Pattern Matching | Derivatives / Primes | Missing Infos | Untranslatable Macro | Explanation | Evaluation Data |
| Gold 1 | \begin{align}J_{-(m+\frac{1}{2})}(x) &= (-1)^{m+1} Y_{m+\frac{1}{2}}(x), \\Y_{-(m+\frac{1}{2})}(x) &= (-1)^m J_{m+\frac{1}{2}}(x).\end{align}
|
Bessel function | 
|
|
- | - | - | - | - | - | Full data:
{
"id": 1,
"pid": 51,
"eid": "math.51.18",
"title": "Bessel function",
"formulae": [
{
"id": "FORMULA_0f521573a47e7fd187dafed615b0ecce",
"formula": "\\begin{align}J_{-(m+\\frac{1}{2})}(x) &= (-1)^{m+1} Y_{m+\\frac{1}{2}}(x), \\\\Y_{-(m+\\frac{1}{2})}(x) &= (-1)^m J_{m+\\frac{1}{2}}(x).\\end{align}",
"semanticFormula": "\\begin{align}\\BesselJ{- (m + \\frac{1}{2})}@{x} &= (- 1)^{m+1} \\BesselY{m+\\frac{1}{2}}@{x} , \\\\ \\BesselY{- (m + \\frac{1}{2})}@{x} &= (-1)^m \\BesselJ{m+\\frac{1}{2}}@{x} .\\end{align}",
"confidence": 0.8803349492974287,
"translations": {
"Mathematica": {
"translation": "BesselJ[- (m +Divide[1,2]), x] == (- 1)^(m + 1)* BesselY[m +Divide[1,2], x]\nBesselY[- (m +Divide[1,2]), x] == (- 1)^(m)* BesselJ[m +Divide[1,2], x]",
"translationInformation": {
"subEquations": [
"BesselJ[- (m +Divide[1,2]), x] = (- 1)^(m + 1)* BesselY[m +Divide[1,2], x]",
"BesselY[- (m +Divide[1,2]), x] = (- 1)^(m)* BesselJ[m +Divide[1,2], x]"
],
"freeVariables": [
"m",
"x"
],
"constraints": [],
"tokenTranslations": {
"\\pgcd": "Greatest common divisor; Example: \\pgcd{m,n}\nWill be translated to: GCD[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/27.1#p2.t1.r3\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/GCD.html",
"\\BesselY": "Bessel function second kind; Example: \\BesselY{v}@{z}\nWill be translated to: BesselY[$0, $1]\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/10.2#E3\nMathematica: https:\/\/",
"\\BesselJ": "Bessel function first kind; Example: \\BesselJ{v}@{z}\nWill be translated to: BesselJ[$0, $1]\nBranch Cuts: if v \\notin \\Integers: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/10.2#E2\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/BesselJ.html"
}
}
},
"Maple": {
"translation": "BesselJ(- (m +(1)\/(2)), x) = (- 1)^(m + 1)* BesselY(m +(1)\/(2), x); BesselY(- (m +(1)\/(2)), x) = (- 1)^(m)* BesselJ(m +(1)\/(2), x)",
"translationInformation": {
"subEquations": [
"BesselJ(- (m +(1)\/(2)), x) = (- 1)^(m + 1)* BesselY(m +(1)\/(2), x)",
"BesselY(- (m +(1)\/(2)), x) = (- 1)^(m)* BesselJ(m +(1)\/(2), x)"
],
"freeVariables": [
"m",
"x"
],
"constraints": [],
"tokenTranslations": {
"\\pgcd": "Greatest common divisor; Example: \\pgcd{m,n}\nWill be translated to: gcd($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/27.1#p2.t1.r3\nMaple: https:\/\/www.maplesoft.com\/support\/help\/Maple\/view.aspx?path=gcd",
"\\BesselY": "Bessel function second kind; Example: \\BesselY{v}@{z}\nWill be translated to: BesselY($0, $1)\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/10.2#E3\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=Bessel",
"\\BesselJ": "Bessel function first kind; Example: \\BesselJ{v}@{z}\nWill be translated to: BesselJ($0, $1)\nBranch Cuts: if v \\notin \\Integers: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/10.2#E2\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=Bessel"
}
}
}
},
"positions": [
{
"section": 8,
"sentence": 8,
"word": 32
}
],
"includes": [
"Y_{\\alpha}",
"J_{-\\alpha}(x)",
"J",
"J_{\\alpha}(x)",
"Y_{n}",
"J_{n}(x)",
"m",
"Y_{\\alpha}(x)",
"J_{\\alpha}",
"x",
"(-1)^{m}",
"J_{n}",
"J_{\\alpha}(z)",
"J_{\\alpha}(k)",
"Y",
"J_{n + m}(x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Bessel function first kind",
"score": 2
},
{
"definition": "Bessel function second kind",
"score": 2
},
{
"definition": "above relation",
"score": 0
},
{
"definition": "spherical Bessel",
"score": 1
},
{
"definition": "integer",
"score": 1
},
{
"definition": "nonnegative integer",
"score": 1
},
{
"definition": "relationship",
"score": 0
},
{
"definition": "function",
"score": 1
},
{
"definition": "recurrence relation",
"score": 1
},
{
"definition": "Bessel",
"score": 1
},
{
"definition": "large number of other known integral",
"score": 0
},
{
"definition": "positive zero",
"score": 0
},
{
"definition": "entire function of genus",
"score": 0
},
{
"definition": "identity",
"score": 0
},
{
"definition": "orthogonality relation",
"score": 0
},
{
"definition": "Bessel function",
"score": 2
},
{
"definition": "term",
"score": 0
},
{
"definition": "real zero",
"score": 0
},
{
"definition": "similar relation",
"score": 0
},
{
"definition": "Hankel",
"score": 1
},
{
"definition": "Bessel function of the second kind",
"score": 2
},
{
"definition": "limit",
"score": 0
},
{
"definition": "ordinary Bessel function",
"score": 1
},
{
"definition": "case",
"score": 0
},
{
"definition": "negative integer",
"score": 0
},
{
"definition": "integral formula",
"score": 0
},
{
"definition": "small argument",
"score": 0
},
{
"definition": "average",
"score": 0
},
{
"definition": "Bessel function of the first kind",
"score": 2
},
{
"definition": "reference",
"score": 0
},
{
"definition": "series expansion",
"score": 0
},
{
"definition": "spherical Bessel function",
"score": 1
},
{
"definition": "Abel 's identity",
"score": 0
},
{
"definition": "important property of Bessel 's equation",
"score": 1
},
{
"definition": "particular Bessel",
"score": 1
},
{
"definition": "solution of Bessel 's equation",
"score": 0
},
{
"definition": "Wronskian of the solution",
"score": 0
},
{
"definition": "series",
"score": 0
},
{
"definition": "closure equation",
"score": 0
}
]
}
]
}
|
| Gold 2 | E(e) \,=\, \int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta
|
Ellipse | 
|
|
- | 
|
- | - | - | e was interpreted as Euler's number
|
Full data:
{
"id": 2,
"pid": 52,
"eid": "math.52.404",
"title": "Ellipse",
"formulae": [
{
"id": "FORMULA_d3e28ddd096754fb8e1e52aaaa4f7770",
"formula": "E(e) \\,=\\, \\int_0^{\\pi\/2}\\sqrt {1 - e^2 \\sin^2\\theta}\\ d\\theta",
"semanticFormula": "\\compellintEk@{e} = \\int_0^{\\cpi \/ 2} \\sqrt{1 - e^2 \\sin^2 \\theta} \\diff{\\theta}",
"confidence": 0.8896531556938116,
"translations": {
"Mathematica": {
"translation": "EllipticE[(e)^2] == Integrate[Sqrt[1 - (e)^(2)*(Sin[\\[Theta]])^(2)], {\\[Theta], 0, Pi\/2}, GenerateConditions->None]",
"translationInformation": {
"subEquations": [
"EllipticE[(e)^2] = Integrate[Sqrt[1 - (e)^(2)*(Sin[\\[Theta]])^(2)], {\\[Theta], 0, Pi\/2}, GenerateConditions->None]"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\cpi": "Pi was translated to: Pi",
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\compellintEk": "Legendre's complete elliptic integral of the second kind; Example: \\compellintEk@{k}\nWill be translated to: EllipticE[($0)^2]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/19.2#E8\nMathematica: https:\/\/",
"\\sin": "Sine; Example: \\sin@@{z}\nWill be translated to: Sin[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E1\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Sin.html"
}
}
},
"Maple": {
"translation": "EllipticE(e) = int(sqrt(1 - (e)^(2)*(sin(theta))^(2)), theta = 0..Pi\/2)",
"translationInformation": {
"subEquations": [
"EllipticE(e) = int(sqrt(1 - (e)^(2)*(sin(theta))^(2)), theta = 0..Pi\/2)"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\cpi": "Pi was translated to: Pi",
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\compellintEk": "Legendre's complete elliptic integral of the second kind; Example: \\compellintEk@{k}\nWill be translated to: EllipticE($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/19.2#E8\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=EllipticE",
"\\sin": "Sine; Example: \\sin@@{z}\nWill be translated to: sin($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E1\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=sin"
}
}
}
},
"positions": [
{
"section": 37,
"sentence": 0,
"word": 39
}
],
"includes": [
"\\theta",
"E",
"\\pi a b",
"\\pi",
"e",
"E(e)"
],
"isPartOf": [],
"definiens": [
{
"definition": "complete elliptic integral of the second kind",
"score": 2
},
{
"definition": "elementary function",
"score": 1
},
{
"definition": "function",
"score": 1
},
{
"definition": "length of the semi-major axis",
"score": 2
},
{
"definition": "eccentricity",
"score": 2
},
{
"definition": "circumference",
"score": 0
},
{
"definition": "ellipse",
"score": 1
},
{
"definition": "angle",
"score": 1
},
{
"definition": "angular coordinate",
"score": 1
},
{
"definition": "center",
"score": 0
},
{
"definition": "formula",
"score": 0
},
{
"definition": "rotation angle",
"score": 0
}
]
}
]
}
|
| Gold 3 | F(x;k) = u
|
Elliptic integral |
|
|
|
- | - | - | - | x is substituted
|
Full data:
{
"id": 3,
"pid": 53,
"eid": "math.53.6",
"title": "Elliptic integral",
"formulae": [
{
"id": "FORMULA_04e9de23897a3b23dee1a9b7312ad99e",
"formula": "F(x;k) = u",
"semanticFormula": "\\incellintFk@{\\asin@{\\Jacobiellsnk@@{u}{k}}}{k} = u",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "EllipticF[ArcSin[JacobiSN[u, (k)^2]], (k)^2] == u",
"translationInformation": {
"subEquations": [
"EllipticF[ArcSin[JacobiSN[u, (k)^2]], (k)^2] = u"
],
"freeVariables": [
"k",
"u"
],
"constraints": [],
"tokenTranslations": {}
}
},
"Maple": {
"translation": "EllipticF(JacobiSN(u, k), k) = u",
"translationInformation": {
"subEquations": [
"EllipticF(JacobiSN(u, k), k) = u"
],
"freeVariables": [
"k",
"u"
],
"constraints": [],
"tokenTranslations": {}
}
}
},
"positions": [
{
"section": 2,
"sentence": 6,
"word": 5
}
],
"includes": [
"u",
"F",
"x",
"k"
],
"isPartOf": [
"F(\\varphi,k) = F\\left(\\varphi \\,|\\, k^2\\right) = F(\\sin \\varphi ; k) = \\int_0^\\varphi \\frac {\\mathrm{d}\\theta}{\\sqrt{1 - k^2 \\sin^2 \\theta}}",
"F(x ; k) = \\int_{0}^{x} \\frac{\\mathrm{d}t}{\\sqrt{\\left(1 - t^2\\right)\\left(1 - k^2 t^2\\right)}}",
"E(\\varphi,k) = E\\left(\\varphi \\,|\\,k^2\\right) = E(\\sin\\varphi;k) = \\int_0^\\varphi \\sqrt{1-k^2 \\sin^2\\theta}\\,\\mathrm{d}\\theta",
"E(x;k) = \\int_0^x \\frac{\\sqrt{1-k^2 t^2} }{\\sqrt{1-t^2}}\\,\\mathrm{d}t"
],
"definiens": [
{
"definition": "inverse to the elliptic integral",
"score": 1
},
{
"definition": "Jacobian elliptic function",
"score": 2
},
{
"definition": "Legendre",
"score": 1
},
{
"definition": "normal form",
"score": 1
},
{
"definition": "trigonometric form",
"score": 1
},
{
"definition": "incomplete elliptic integral of the second kind",
"score": 0
},
{
"definition": "incomplete elliptic integral of the first kind",
"score": 2
}
]
}
]
}
|
| Gold 4 | \frac{1}{\Gamma(z)} = \frac{i}{2\pi}\int_C (-t)^{-z}e^{-t}\,dt
|
Gamma function |
|
- | - |
|
- | - | - | Contour integrals cannot be translated. | Full data:
{
"id": 4,
"pid": 54,
"eid": "math.54.195",
"title": "Gamma function",
"formulae": [
{
"id": "FORMULA_19a0f00da77cc439ad679c579a295bd2",
"formula": "\\frac{1}{\\Gamma(z)} = \\frac{i}{2\\pi}\\int_C (-t)^{-z}e^{-t}\\,dt",
"semanticFormula": "\\frac{1}{\\EulerGamma@{z}} = \\frac{\\iunit}{2 \\cpi} \\int_C(- t)^{-z} \\expe^{-t} \\diff{t}",
"confidence": 0.8809245132365588,
"translations": {},
"positions": [
{
"section": 11,
"sentence": 10,
"word": 9
}
],
"includes": [
"C",
"\\Gamma",
"\\frac {1}{\\Gamma (z)}",
"z",
"1",
"\\Gamma(r)",
"t",
"\\pi",
"\\Gamma (z)",
"\\Gamma(z)",
"\\Pi\\left(z\\right)",
"\\Gamma\\left(z\\right)",
"e^{-x}"
],
"isPartOf": [],
"definiens": [
{
"definition": "related expression",
"score": 0
},
{
"definition": "integer",
"score": 0
},
{
"definition": "reflection formula",
"score": 1
},
{
"definition": "end",
"score": 0
},
{
"definition": "Hankel contour",
"score": 2
},
{
"definition": "Riemann sphere",
"score": 1
},
{
"definition": "Hankel 's formula for the gamma function",
"score": 2
},
{
"definition": "gamma function",
"score": 2
},
{
"definition": "reciprocal gamma function",
"score": 2
}
]
}
]
}
|
| Gold 5 | 2^{4} = 2 \times2 \times 2 \times 2 = 16
|
Logarithm |
|
|
- | - | - | - | - | - | Full data:
{
"id": 5,
"pid": 55,
"eid": "",
"title": "Logarithm",
"formulae": [
{
"id": "FORMULA_579837194f2124b255d579031524a91c",
"formula": "2^{4} = 2 \\times2 \\times 2 \\times 2 = 16",
"semanticFormula": "2^{4} = 2 \\times2 \\times 2 \\times 2 = 16",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "(2)^(4) == 2 * 2 * 2 * 2 == 16",
"translationInformation": {
"subEquations": [
"(2)^(4) = 2 * 2 * 2 * 2",
"2 * 2 * 2 * 2 = 16"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\times": "was translated to: *"
}
}
},
"Maple": {
"translation": "(2)^(4) = 2 * 2 * 2 * 2 = 16",
"translationInformation": {
"subEquations": [
"(2)^(4) = 2 * 2 * 2 * 2",
"2 * 2 * 2 * 2 = 16"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\times": "was translated to: *"
}
}
}
},
"positions": [
{
"section": 4,
"sentence": 0,
"word": 3
}
],
"includes": [
"2",
"^{4}"
],
"isPartOf": [],
"definiens": [
{
"definition": "example",
"score": 2
}
]
}
]
}
|
| Gold 6 | \psi(x) := \sum_{n=1}^\infty e^{-n^2 \pi x}
|
Riemann zeta function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 6,
"pid": 56,
"eid": "math.56.40",
"title": "Riemann zeta function",
"formulae": [
{
"id": "FORMULA_bd88ec58aa42c7a59bc2f4ff458a54cf",
"formula": "\\psi(x) := \\sum_{n=1}^\\infty e^{-n^2 \\pi x}",
"semanticFormula": "\\psi(x) : = \\sum_{n=1}^\\infty \\expe^{- n^2 \\cpi x}",
"confidence": 0.9073333333333333,
"translations": {
"Mathematica": {
"translation": "\\[Psi][x_] := Sum[Exp[-(n)^(2)*Pi*x], {n, 1, Infinity}]"
},
"Maple": {
"translation": "psi := (x) -> sum(exp(-(n)^(2)*Pi*x), n=1..infinity)"
}
},
"positions": [
{
"section": 4,
"sentence": 7,
"word": 23
}
],
"includes": [
"1",
"n",
"2",
"x",
"\\psi"
],
"isPartOf": [],
"definiens": [
{
"definition": "analytic continuation",
"score": 0
},
{
"definition": "absolute convergence",
"score": 0
},
{
"definition": "convenience",
"score": 0
},
{
"definition": "inversion",
"score": 0
},
{
"definition": "process",
"score": 0
},
{
"definition": "stricter requirement",
"score": 0
},
{
"definition": "series",
"score": 1
},
{
"definition": "definition",
"score": 2
}
]
}
]
}
|
| Gold 7 | \operatorname{li}(x) = \lim_{\varepsilon \to 0+} \left( \int_0^{1-\varepsilon} \frac{dt}{\ln t} + \int_{1+\varepsilon}^x \frac{dt}{\ln t} \right)
|
Logarithmic integral function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 7,
"pid": 57,
"eid": "math.57.2",
"title": "Logarithmic integral function",
"formulae": [
{
"id": "FORMULA_36fb8f8330168b8f8acda0dc36851474",
"formula": "\\operatorname{li}(x) = \\lim_{\\varepsilon \\to 0+} \\left( \\int_0^{1-\\varepsilon} \\frac{dt}{\\ln t} + \\int_{1+\\varepsilon}^x \\frac{dt}{\\ln t} \\right)",
"semanticFormula": "\\logint@{x} = \\lim_{\\varepsilon \\to 0+}(\\int_0^{1-\\varepsilon} \\frac{\\diff{t}}{\\ln t} + \\int_{1+\\varepsilon}^x \\frac{\\diff{t}}{\\ln t})",
"confidence": 0.8728566391293461,
"translations": {
"Mathematica": {
"translation": "LogIntegral[x] == Limit[Integrate[Divide[1,Log[t]], {t, 0, 1 - \\[CurlyEpsilon]}, GenerateConditions->None]+ Integrate[Divide[1,Log[t]], {t, 1 + \\[CurlyEpsilon], x}, GenerateConditions->None], \\[CurlyEpsilon] -> 0, Direction -> \"FromAbove\", GenerateConditions->None]",
"translationInformation": {
"subEquations": [
"LogIntegral[x] = Limit[Integrate[Divide[1,Log[t]], {t, 0, 1 - \\[CurlyEpsilon]}, GenerateConditions->None]+ Integrate[Divide[1,Log[t]], {t, 1 + \\[CurlyEpsilon], x}, GenerateConditions->None], \\[CurlyEpsilon] -> 0, Direction -> \"FromAbove\", GenerateConditions->None]"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\logint": "Logarithmic integral; Example: \\logint@{x}\nWill be translated to: LogIntegral[$0]\nConstraints: x > 1\nMathematica uses other branch cuts: (-\\inf, 1)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/6.2#E8\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/LogIntegral.html",
"\\ln": "Natural logarithm; Example: \\ln@@{z}\nWill be translated to: Log[$0]\nConstraints: z != 0\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.2#E2\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Log.html"
}
}
},
"Maple": {
"translation": "Li(x) = limit(int((1)\/(ln(t)), t = 0..1 - varepsilon)+ int((1)\/(ln(t)), t = 1 + varepsilon..x), varepsilon = 0, right)",
"translationInformation": {
"subEquations": [
"Li(x) = limit(int((1)\/(ln(t)), t = 0..1 - varepsilon)+ int((1)\/(ln(t)), t = 1 + varepsilon..x), varepsilon = 0, right)"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\logint": "Logarithmic integral; Example: \\logint@{x}\nWill be translated to: Li($0)\nConstraints: x > 1\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/6.2#E8\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=Li",
"\\ln": "Natural logarithm; Example: \\ln@@{z}\nWill be translated to: ln($0)\nConstraints: z != 0\nBranch Cuts: (-\\infty, 0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.2#E2\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=ln"
}
}
}
},
"positions": [
{
"section": 1,
"sentence": 2,
"word": 22
}
],
"includes": [
"x",
"x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Cauchy principal value",
"score": 2
},
{
"definition": "function",
"score": 1
},
{
"definition": "singularity",
"score": 1
},
{
"definition": "special function",
"score": 1
},
{
"definition": "integral representation",
"score": 1
},
{
"definition": "integral logarithm li",
"score": 2
},
{
"definition": "logarithmic integral function",
"score": 2
},
{
"definition": "logarithmic integral",
"score": 2
},
{
"definition": "function li",
"score": 1
}
]
}
]
}
|
| Gold 8 | w_{i} = \frac{1}{p'_{n}(x_{i})}\int_{a}^{b}\omega(x)\frac{p_{n}(x)}{x-x_{i}}dx
|
Gaussian quadrature |
|
|
- | - |
|
- | - | - | Full data:
{
"id": 8,
"pid": 58,
"eid": "math.58.61",
"title": "Gaussian quadrature",
"formulae": [
{
"id": "FORMULA_8c49145544fca24efb8de07eb1275c09",
"formula": "w_{i} = \\frac{1}{p'_{n}(x_{i})}\\int_{a}^{b}\\omega(x)\\frac{p_{n}(x)}{x-x_{i}}dx",
"semanticFormula": "w_{i} = \\frac{1}{p'_{n}(x_{i})} \\int_{a}^{b} \\omega(x) \\frac{p_{n}(x)}{x-x_{i}} \\diff{x}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Subscript[w, i] = Divide[1, Subscript[p\\[Prime], n][Subscript[x, i]]]*Integrate[\\[Omega][x]*Divide[Subscript[p,n][x], x-Subscript[x,i]], {x, a, b}]"
}
},
"positions": [
{
"section": 5,
"sentence": 4,
"word": 24
}
],
"includes": [
"a",
"b",
"w_{i}",
"p_n(x)",
"p_{k}(x)",
"p_{n}",
"x_{i}",
"\\omega(x)",
"p_{n}(x)",
"\\omega",
"x_i",
"a_{n}",
"P_{n}",
"w_i",
"r(x_{i})",
"i",
"n",
"x",
"P_{n}(x)",
"\\frac{p_{n}(x)}{x-x_{i}}",
"p'_{n}(x_{i})",
"p_{n}(x_{i})",
"x_{j}",
"p_r",
"p_s",
"\\mathbf{e}_n",
"x_j",
"1"
],
"isPartOf": [],
"definiens": [
{
"definition": "yield",
"score": 0
},
{
"definition": "integral expression for the weight",
"score": 2
},
{
"definition": "integrand",
"score": 1
},
{
"definition": "L'H\u00f4pital 's rule",
"score": 0
},
{
"definition": "limit",
"score": 0
},
{
"definition": "polynomial of degree",
"score": 0
}
]
}
]
}
|
| Gold 9 | \begin{align}x & =ue^u, \\[5pt]\frac{dx}{du} & =(u+1)e^u.\end{align}
|
Lambert W function |
|
|
|
- | - | - | - | - | Full data:
{
"id": 9,
"pid": 59,
"eid": "math.59.52",
"title": "Lambert W function",
"formulae": [
{
"id": "FORMULA_fe13643d8449f601f150fd50c0751cf2",
"formula": "\\begin{align}x & =ue^u, \\\\[5pt]\\frac{dx}{du} & =(u+1)e^u.\\end{align}",
"semanticFormula": "\\begin{align}x & =\\LambertW@{x}\\expe^{\\LambertW@{x}}, \\\\ \\deriv{x}{\\LambertW@{x}} &=(\\LambertW@{x} + 1) \\expe^{\\LambertW@{x}} .\\end{align}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "x == ProductLog[x]*(E)^(ProductLog[x])\nD[x,ProductLog[x]] = (ProductLog[x] + 1)*Exp[ProductLog[x]]",
"translationInformation": {
"subEquations": [
"x = ProductLog[x]*(E)^(ProductLog[x])",
"D[x,ProductLog[x]] = (ProductLog[x] + 1)*Exp[ProductLog[x]]"
],
"freeVariables": [
"u",
"x"
],
"constraints": [],
"tokenTranslations": {
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function."
}
}
},
"Maple": {
"translation": "x = LambertW(x)*exp(u); diff(x, [LambertW(x)$1]) = (LambertW(x) + 1)*exp(LambertW(x))",
"translationInformation": {
"subEquations": [
"x = LambertW(x)*exp(u)",
"diff(x, [LambertW(x)$1]) = (LambertW(x) + 1)*exp(LambertW(x))"
],
"freeVariables": [
"u",
"x"
],
"constraints": [],
"tokenTranslations": {
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function."
}
}
}
},
"positions": [
{
"section": 12,
"sentence": 1,
"word": 14
}
],
"includes": [
"e^{w}",
"x"
],
"isPartOf": [],
"definiens": [
{
"definition": "substitution",
"score": 2
},
{
"definition": "third identity",
"score": 0
},
{
"definition": "second identity",
"score": 1
}
]
}
]
}
|
| Gold 10 | \frac{1}{\left| \mathbf{x}-\mathbf{x}' \right|} = \frac{1}{\sqrt{r^2+{r'}^2-2r{r'}\cos\gamma}} = \sum_{\ell=0}^\infty \frac{{r'}^\ell}{r^{\ell+1}} P_\ell(\cos \gamma)
|
Legendre polynomials |
|
|
- | - |
|
- | - | - | Full data:
{
"id": 10,
"pid": 60,
"eid": "math.60.57",
"title": "Legendre polynomials",
"formulae": [
{
"id": "FORMULA_8646bd0d06e9454aaa39dfc506fe54f7",
"formula": "\\frac{1}{\\left| \\mathbf{x}-\\mathbf{x}' \\right|} = \\frac{1}{\\sqrt{r^2+{r'}^2-2r{r'}\\cos\\gamma}} = \\sum_{\\ell=0}^\\infty \\frac{{r'}^\\ell}{r^{\\ell+1}} P_\\ell(\\cos \\gamma)",
"semanticFormula": "\\frac{1}{|\\mathbf{x} - \\mathbf{x} '|} = \\frac{1}{\\sqrt{r^2+{r'}^2-2r{r'}\\cos\\gamma}} = \\sum_{\\ell=0}^\\infty \\frac{{r'}^\\ell}{r^{\\ell+1}} \\LegendrepolyP{\\ell}@{\\cos \\gamma}",
"confidence": 0.808438593520797,
"translations": {
"Mathematica": "Divide[1, Abs[x - x\\[Prime]]] == Divide[1, Sqrt[r^2+(r\\[Prime])^(2)-2*r*r\\[Prime] Cos[\\[Gamma]]]] == Sum[Divide[(r\\[Prime])^(\\[ScriptL]), r^(\\[ScriptL]+1)]*LegendreP[\\[ScriptL], Cos[\\[Gamma]]], {\\[ScriptL], 0, Infinity}]"
},
"positions": [
{
"section": 6,
"sentence": 0,
"word": 21
}
],
"includes": [
"P_n(x)",
"P_n",
"P_n(\\cos\\theta)",
"P_{n}(x)",
"P_m",
"r",
"r{'}",
"\\mathbf{x}",
"\\mathbf{x}{'}",
"\\gamma",
"P"
],
"isPartOf": [],
"definiens": [
{
"definition": "expansion",
"score": 2
},
{
"definition": "Adrien-Marie Legendre as the coefficient",
"score": 0
},
{
"definition": "angle",
"score": 1
},
{
"definition": "Legendre polynomial",
"score": 2
},
{
"definition": "length of the vector",
"score": 1
},
{
"definition": "vector",
"score": 1
},
{
"definition": "polynomial",
"score": 1
}
]
}
]
}
|
| Gold 11 | \operatorname{erf}^{(k)}(z) = \frac{2 (-1)^{k-1}}{\sqrt{\pi}} \mathit{H}_{k-1}(z) e^{-z^2} = \frac{2}{\sqrt{\pi}} \frac{d^{k-1}}{dz^{k-1}} \left(e^{-z^2}\right),\qquad k=1, 2, \dots
|
Error function |
|
|
- | - |
|
- | - | was not detected as k-th derivative but as power. | Full data:
{
"id": 11,
"pid": 61,
"eid": "math.61.27",
"title": "Error function",
"formulae": [
{
"id": "FORMULA_523ec091d0929f0fa69ae7e0d563a72b",
"formula": "\\operatorname{erf}^{(k)}(z) = \\frac{2 (-1)^{k-1}}{\\sqrt{\\pi}} \\mathit{H}_{k-1}(z) e^{-z^2} = \\frac{2}{\\sqrt{\\pi}} \\frac{d^{k-1}}{dz^{k-1}} \\left(e^{-z^2}\\right),\\qquad k=1, 2, \\dots",
"semanticFormula": "\\erf@@{(z)}^{(k)} = \\frac{2 (-1)^{k-1}}{\\sqrt{\\cpi}} \\HermitepolyH{k-1}@{z} \\expe^{-z^2} = \\frac{2}{\\sqrt{\\cpi}} \\deriv [{k-1}]{ }{z}(\\expe^{-z^2}) , \\qquad k = 1 , 2 , \\dots",
"confidence": 0.82607945540953,
"translations": {
"Mathematica": {
"translation": "D[Erf[z], {z, k}] == Divide[2*(- 1)^(k - 1),Sqrt[Pi]]*HermiteH[k - 1, z]*Exp[- (z)^(2)] == Divide[2,Sqrt[Pi]]*D[Exp[- (z)^(2)], {z, k - 1}]",
"translationInformation": {
"subEquations": [
"D[Erf[z], {z, k}] = Divide[2*(- 1)^(k - 1),Sqrt[Pi]]*HermiteH[k - 1, z]*Exp[- (z)^(2)]",
"Divide[2*(- 1)^(k - 1),Sqrt[Pi]]*HermiteH[k - 1, z]*Exp[- (z)^(2)] = Divide[2,Sqrt[Pi]]*D[Exp[- (z)^(2)], {z, k - 1}]"
],
"freeVariables": [
"k",
"z"
],
"constraints": [
"k == 1 , 2 , \\[Ellipsis]"
],
"tokenTranslations": {
"\\deriv1": "Derivative; Example: \\deriv[n]{f}{x}\nWill be translated to: D[$1, {$2, $0}]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/1.4#E4\nMathematica: https:\/\/",
"\\cpi": "Pi was translated to: Pi",
"\\HermitepolyH": "Hermite polynomial; Example: \\HermitepolyH{n}@{x}\nWill be translated to: HermiteH[$0, $1]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/18.3#T1.t1.r13\nMathematica: https:\/\/",
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\erf": "Error function; Example: \\erf@@{z}\nWill be translated to: Erf[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/7.2#E1\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Erf.html"
}
}
},
"Maple": {
"translation": "diff(erf(z), [z$k]) = (2*(- 1)^(k - 1))\/(sqrt(Pi))*HermiteH(k - 1, z)*exp(- (z)^(2)) = (2)\/(sqrt(Pi))*diff(exp(- (z)^(2)), [z$(k - 1)])",
"translationInformation": {
"subEquations": [
"diff(erf(z), [z$k]) = (2*(- 1)^(k - 1))\/(sqrt(Pi))*HermiteH(k - 1, z)*exp(- (z)^(2))",
"(2*(- 1)^(k - 1))\/(sqrt(Pi))*HermiteH(k - 1, z)*exp(- (z)^(2)) = (2)\/(sqrt(Pi))*diff(exp(- (z)^(2)), [z$(k - 1)])"
],
"freeVariables": [
"k",
"z"
],
"constraints": [
"k = 1 , 2 , .."
],
"tokenTranslations": {
"\\deriv1": "Derivative; Example: \\deriv[n]{f}{x}\nWill be translated to: diff($1, [$2$($0)])\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/1.4#E4\nMaple: https:\/\/www.maplesoft.com\/support\/help\/Maple\/view.aspx?path=diff",
"\\cpi": "Pi was translated to: Pi",
"\\HermitepolyH": "Hermite polynomial; Example: \\HermitepolyH{n}@{x}\nWill be translated to: HermiteH($0, $1)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/18.3#T1.t1.r13\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=HermiteH",
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\erf": "Error function; Example: \\erf@@{z}\nWill be translated to: erf($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/7.2#E1\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=erf"
}
}
}
},
"positions": [
{
"section": 5,
"sentence": 4,
"word": 6
}
],
"includes": [
"erf",
"e^{-t^2}",
"-1",
"z",
"z)",
"e",
"\\mathit{H}",
"z^{\\bar{n}}"
],
"isPartOf": [],
"definiens": [
{
"definition": "Higher order derivative",
"score": 2
},
{
"definition": "physicists ' Hermite polynomial",
"score": 1
},
{
"definition": "name error function",
"score": 1
},
{
"definition": "erfc",
"score": 1
},
{
"definition": "error function",
"score": 2
},
{
"definition": "erf",
"score": 1
}
]
}
]
}
|
| Gold 12 | x_k = \cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1
|
Chebyshev polynomials |
|
|
- | - | - | - | - | - | Full data:
{
"id": 12,
"pid": 62,
"eid": "math.62.44",
"title": "Chebyshev polynomials",
"formulae": [
{
"id": "FORMULA_d9eb68704833b0f525c4ca81a749d9ca",
"formula": "x_k = \\cos\\left(\\frac{\\pi(k+1\/2)}{n}\\right),\\quad k=0,\\ldots,n-1",
"semanticFormula": "x_k = \\cos(\\frac{\\cpi(k + 1 \/ 2)}{n}) , \\quad k = 0 , \\ldots , n - 1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Subscript[x, k] == Cos[Divide[Pi*(k + 1\/2),n]]",
"translationInformation": {
"subEquations": [
"Subscript[x, k] = Cos[Divide[Pi*(k + 1\/2),n]]"
],
"freeVariables": [
"Subscript[x, k]",
"k",
"n"
],
"constraints": [
"k == 0 , \\[Ellipsis], n - 1"
],
"tokenTranslations": {
"\\cos": "Cosine; Example: \\cos@@{z}\nWill be translated to: Cos[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E2\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Cos.html",
"\\cpi": "Pi was translated to: Pi"
}
}
},
"Maple": {
"translation": "x[k] = cos((Pi*(k + 1\/2))\/(n))",
"translationInformation": {
"subEquations": [
"x[k] = cos((Pi*(k + 1\/2))\/(n))"
],
"freeVariables": [
"k",
"n",
"x[k]"
],
"constraints": [
"k = 0 , .. , n - 1"
],
"tokenTranslations": {
"\\cos": "Cosine; Example: \\cos@@{z}\nWill be translated to: cos($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E2\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=cos",
"\\cpi": "Pi was translated to: Pi"
}
}
}
},
"positions": [
{
"section": 8,
"sentence": 2,
"word": 18
}
],
"includes": [
"n",
"x",
"n x",
"-1",
"k = 0",
"x_{k}"
],
"isPartOf": [],
"definiens": [
{
"definition": "root",
"score": 2
},
{
"definition": "one",
"score": 0
},
{
"definition": "trigonometric definition",
"score": 0
},
{
"definition": "fact",
"score": 0
},
{
"definition": "different simple root",
"score": 1
},
{
"definition": "Chebyshev polynomial of the first kind",
"score": 1
},
{
"definition": "Chebyshev polynomial",
"score": 1
}
]
}
]
}
|
| Gold 13 | E(x, y; u) := \sum_{n=0}^\infty u^n \, \psi_n (x) \, \psi_n (y) = \frac{1}{\sqrt{\pi (1 - u^2)}} \, \exp\left(-\frac{1 - u}{1 + u} \, \frac{(x + y)^2}{4} - \frac{1 + u}{1 - u} \, \frac{(x - y)^2}{4}\right)
|
Hermite polynomials |
|
|
- | - | - | - | - | - | Full data:
{
"id": 13,
"pid": 63,
"eid": "math.63.109",
"title": "Hermite polynomials",
"formulae": [
{
"id": "FORMULA_249043719eb4dd70350b460363255e11",
"formula": "E(x, y; u) := \\sum_{n=0}^\\infty u^n \\, \\psi_n (x) \\, \\psi_n (y) = \\frac{1}{\\sqrt{\\pi (1 - u^2)}} \\, \\exp\\left(-\\frac{1 - u}{1 + u} \\, \\frac{(x + y)^2}{4} - \\frac{1 + u}{1 - u} \\, \\frac{(x - y)^2}{4}\\right)",
"semanticFormula": "E(x , y ; u) : = \\sum_{n=0}^\\infty u^n \\psi_n(x) \\psi_n(y) = \\frac{1}{\\sqrt{\\cpi(1 - u^2)}} \\exp(- \\frac{1 - u}{1 + u} \\frac{(x + y)^2}{4} - \\frac{1 + u}{1 - u} \\frac{(x - y)^2}{4})",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "\\[CapitalEpsilon][x_, y_, u_] := Sum[(u)^(n)* Subscript[\\[Psi], n][x]* Subscript[\\[Psi], n][y], {n, 0, Infinity}, GenerateConditions->None] == Divide[1,Sqrt[Pi*(1 - (u)^(2))]]*Exp[-Divide[1 - u,1 + u]*Divide[(x + y)^(2),4]-Divide[1 + u,1 - u]*Divide[(x - y)^(2),4]]"
},
"Maple": {
"translation": "Epsilon := (x, y, u) -> sum((u)^(n)* psi[n](x)* psi[n](y), n = 0..infinity) = (1)\/(sqrt(Pi*(1 - (u)^(2))))*exp(-(1 - u)\/(1 + u)*((x + y)^(2))\/(4)-(1 + u)\/(1 - u)*((x - y)^(2))\/(4))"
}
},
"positions": [
{
"section": 25,
"sentence": 2,
"word": 16
}
],
"includes": [
"u",
"\\psi_{n}",
"H_{n}(x)",
"\\psi_{n}(x)",
"x^{n}",
"n",
"x",
"H_{n}",
"He_{n}(x)",
"He_{n}",
"D_{n}(z)",
"E(x,y;u)",
"H_{n}(y)"
],
"isPartOf": [],
"definiens": [
{
"definition": "distributional identity",
"score": 1
},
{
"definition": "separable kernel",
"score": 1
},
{
"definition": "Mehler 's formula",
"score": 2
},
{
"definition": "Hermite polynomial",
"score": 1
},
{
"definition": "Hermite function",
"score": 2
},
{
"definition": "Hermite",
"score": 1
},
{
"definition": "bivariate Gaussian probability density",
"score": 1
},
{
"definition": "Gaussian probability density",
"score": 1
},
{
"definition": "Gaussian probability",
"score": 1
}
]
}
]
}
|
| Gold 14 | x = \pm 1
|
Legendre function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 14,
"pid": 64,
"eid": "math.64.8",
"title": "Legendre function",
"formulae": [
{
"id": "FORMULA_06f9b7b1d3f141742ad1c582b55056ba",
"formula": "x = \\pm 1",
"semanticFormula": "x = \\pm 1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "x == \\[PlusMinus]1",
"translationInformation": {
"subEquations": [
"x = + 1",
"x = - 1"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\pm": "was translated to: \\[PlusMinus]"
}
}
},
"Maple": {
"translation": "x = &+- 1",
"translationInformation": {
"subEquations": [
"x = + 1",
"x = - 1"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\pm": "was translated to: &+-"
}
}
}
},
"positions": [
{
"section": 3,
"sentence": 1,
"word": 11
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| Gold 15 | E_n=2^nE_n(\tfrac{1}{2})
|
Bernoulli polynomials |
|
|
- |
|
- | - | - | Both E where detected as Euler's number but the second is Euler polynomial. | Full data:
{
"id": 15,
"pid": 65,
"eid": "math.65.27",
"title": "Bernoulli polynomials",
"formulae": [
{
"id": "FORMULA_a7fcf738c676932d58f39ff9f7df22ae",
"formula": "E_n=2^nE_n(\\tfrac{1}{2})",
"semanticFormula": "\\EulernumberE{n} = 2^n\\EulerpolyE{n}@{\\tfrac{1}{2}}",
"confidence": 0.8953028732079359,
"translations": {
"Mathematica": {
"translation": "EulerE[n] == (2)^(n)* EulerE[n, Divide[1,2]]"
},
"Maple": {
"translation": "euler(n) = (2)^(n)* euler(n, (1)\/(2))"
}
},
"positions": [
{
"section": 8,
"sentence": 4,
"word": 6
}
],
"includes": [
"B_{n}",
"n",
"E_{k}"
],
"isPartOf": [],
"definiens": [
{
"definition": "Euler number",
"score": 2
}
]
}
]
}
|
| Gold 16 | \operatorname{Si}(ix) = i\operatorname{Shi}(x)
|
Trigonometric integral |
|
|
- | - | - | - | - | Integral was not tagged as a noun by CoreNLP. Hence, the macro for hyperbolic sine function was retrieved too late and not considered for replacements. | Full data:
{
"id": 16,
"pid": 66,
"eid": "math.66.8",
"title": "Trigonometric integral",
"formulae": [
{
"id": "FORMULA_0feb8031b89a9707b164163ec50265f0",
"formula": "\\operatorname{Si}(ix) = i\\operatorname{Shi}(x)",
"semanticFormula": "\\sinint@{\\iunit x} = \\iunit \\sinhint@{x}",
"confidence": 0.8811682126384021,
"translations": {
"Mathematica": {
"translation": "SinIntegral[I*x] == I*SinhIntegral[x]",
"translationInformation": {
"subEquations": [
"SinIntegral[I*x] == I*SinhIntegral[x]"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"Shi": "Was interpreted as a function call because of a leading \\operatorname.",
"\\iunit": "Imaginary unit was translated to: I",
"\\sinint": "Sine integral; Example: \\sinint@{z}\nWill be translated to: SinIntegral[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/6.2#E9\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/SinIntegral.html"
}
}
},
"Maple": {
"translation": "Si(I*x) = I*Shi(x)",
"translationInformation": {
"subEquations": [
"Si(I*x) = I*Shi(x)"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"Shi": "Was interpreted as a function call because of a leading \\operatorname.",
"\\iunit": "Imaginary unit was translated to: I",
"\\sinint": "Sine integral; Example: \\sinint@{z}\nWill be translated to: Si($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/6.2#E9\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=Si"
}
}
}
},
"positions": [
{
"section": 3,
"sentence": 1,
"word": 9
}
],
"includes": [
"Si",
"Si(x)",
"x"
],
"isPartOf": [],
"definiens": [
{
"definition": "ordinary sine",
"score": 1
},
{
"definition": "Trigonometric integral",
"score": 2
},
{
"definition": "hyperbolic sine integral",
"score": 2
}
]
}
]
}
|
| Gold 17 | f(z)=\frac{1}{\Beta(x,y)}
|
Beta function |
|
|
|
- | - |
|
- | The original formula contained f(z) but should have been f(x,z). This was fixed in the Wikipedia article after we generated the dataset.
|
Full data:
{
"id": 17,
"pid": 67,
"eid": "math.67.29",
"title": "Beta function",
"formulae": [
{
"id": "FORMULA_5f59825d73d63a9990498edca7222261",
"formula": "f(z)=\\frac{1}{\\Beta(x,y)}",
"semanticFormula": "f(x, y) = \\frac{1}{\\EulerBeta@{x}{y}}",
"confidence": 0.8953028732079359,
"translations": {
"Mathematica": {
"translation": "f[x_, y_] := Divide[1,Beta[x, y]]"
},
"Maple": {
"translation": "f := (x,y) -> (1)\/(Beta(x, y))"
}
},
"positions": [
{
"section": 6,
"sentence": 0,
"word": 12
}
],
"includes": [
"x, y",
"\\Beta",
"y",
"x"
],
"isPartOf": [],
"definiens": [
{
"definition": "function about the form",
"score": 0
},
{
"definition": "reciprocal beta function",
"score": 2
},
{
"definition": "definite integral of trigonometric function",
"score": 1
},
{
"definition": "integral representation",
"score": 0
},
{
"definition": "product",
"score": 0
},
{
"definition": "power",
"score": 0
},
{
"definition": "multiple-angle",
"score": 0
},
{
"definition": "beta function",
"score": 2
}
]
}
]
}
|
| Gold 18 | \begin{align}\int x^m e^{ix^n}\,dx & =\frac{x^{m+1}}{m+1}\,_1F_1\left(\begin{array}{c} \frac{m+1}{n}\\1+\frac{m+1}{n}\end{array}\mid ix^n\right) \\[6px]& =\frac{1}{n} i^\frac{m+1}{n}\gamma\left(\frac{m+1}{n},-ix^n\right),\end{align}
|
Fresnel integral |
|
|
- |
|
- | - | - | Matrix argument of does not exist in the DLMF. | Full data:
{
"id": 18,
"pid": 68,
"eid": "math.68.51",
"title": "Fresnel integral",
"formulae": [
{
"id": "FORMULA_b7dae135f3b04317078f86b595fe7dae",
"formula": "\\begin{align}\\int x^m e^{ix^n}\\,dx & =\\frac{x^{m+1}}{m+1}\\,_1F_1\\left(\\begin{array}{c} \\frac{m+1}{n}\\\\1+\\frac{m+1}{n}\\end{array}\\mid ix^n\\right) \\\\[6px]& =\\frac{1}{n} i^\\frac{m+1}{n}\\gamma\\left(\\frac{m+1}{n},-ix^n\\right),\\end{align}",
"semanticFormula": "\\begin{align}\\int x^m \\exp(\\iunit x^n) \\diff{x} &= \\frac{x^{m+1}}{m+1}\\genhyperF{1}{1}@{\\frac{m+1}{n}}{1+\\frac{m+1}{n}}{\\iunit x^n}\\\\ &=\\frac{1}{n} \\iunit^{(m+1)\/n} \\incgamma@{\\frac{m+1}{n}}{-\\iunit x^n}\\end{align}",
"confidence": 0.869061849326977,
"translations": {
"Mathematica": {
"translation": "Integrate[(x)^(m)* Exp[I*(x)^(n)], x, GenerateConditions->None] == Divide[(x)^(m + 1),m + 1]*HypergeometricPFQ[{Divide[m + 1,n]}, {1 +Divide[m + 1,n]}, I*(x)^(n)] == Divide[1,n]*(I)^((m + 1)\/n)* Gamma[Divide[m + 1,n], 0, - I*(x)^(n)]"
},
"Maple": {
"translation": "int((x)^(m)* exp(I*(x)^(n)), x) = ((x)^(m + 1))\/(m + 1)*hypergeom([(m + 1)\/(n)], [1 +(m + 1)\/(n)], I*(x)^(n)) = (1)\/(n)*(I)^((m + 1)\/n)* GAMMA((m + 1)\/(n))-GAMMA((m + 1)\/(n), - I*(x)^(n))"
}
},
"positions": [
{
"section": 5,
"sentence": 0,
"word": 14
}
],
"includes": [
"dx",
"x"
],
"isPartOf": [],
"definiens": [
{
"definition": "incomplete gamma function",
"score": 2
},
{
"definition": "confluent hypergeometric function",
"score": 2
},
{
"definition": "Fresnel integral",
"score": 1
},
{
"definition": "imaginary part",
"score": 1
}
]
}
]
}
|
| Gold 19 | T_n(x) = \frac{\Gamma(1/2)\sqrt{1-x^2}}{(-2)^n\,\Gamma(n+1/2)} \ \frac{d^n}{dx^n}\left([1-x^2]^{n-1/2}\right)
|
Classical orthogonal polynomials |
|
|
- | - | - |
|
- | No info about Gamma function. | Full data:
{
"id": 19,
"pid": 69,
"eid": "math.69.117",
"title": "Classical orthogonal polynomials",
"formulae": [
{
"id": "FORMULA_725c6b6b645d425d3b385ac2c002da77",
"formula": "T_n(x) = \\frac{\\Gamma(1\/2)\\sqrt{1-x^2}}{(-2)^n\\,\\Gamma(n+1\/2)} \\ \\frac{d^n}{dx^n}\\left([1-x^2]^{n-1\/2}\\right)",
"semanticFormula": "\\ChebyshevpolyT{n}@{x} = \\frac{\\EulerGamma{1\/2}\\sqrt{1-x^2}}{(-2)^n\\EulerGamma{n+1\/2}} \\deriv [n]{ }{x}([1 - x^2]^{n-1\/2})",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "ChebyshevT[n, x] == Divide[Gamma[1\/2]*Sqrt[1 - (x)^(2)],(- 2)^(n)* Gamma[n + 1\/2]]*D[(1 - (x)^(2))^(n - 1\/2), {x, n}]"
},
"Maple": {
"translation": "ChebyshevT(n, x) = (GAMMA(1\/2)*sqrt(1 - (x)^(2)))\/((- 2)^(n)* GAMMA(n + 1\/2))*diff((1 - (x)^(2))^(n - 1\/2), [x$(n)])"
}
},
"positions": [
{
"section": 18,
"sentence": 3,
"word": 4
}
],
"includes": [
"\\ L_n",
"H_n",
"P_{n}",
"n-r",
"n",
"P_{n}(x)",
"-1\/2",
"e_{n}",
"P_n",
"\\lambda_{n}",
"-1",
"+1",
"U_n"
],
"isPartOf": [],
"definiens": [
{
"definition": "Rodrigues ' formula",
"score": 2
},
{
"definition": "orthogonal polynomial",
"score": 1
},
{
"definition": "Chebyshev polynomials of the second kind",
"score": 1
},
{
"definition": "classical orthogonal polynomial",
"score": 1
},
{
"definition": "Chebyshev polynomial",
"score": 2
},
{
"definition": "Gamma function",
"score": 2
}
]
}
]
}
|
| Gold 20 | {}_1F_0(1;;z) = \sum_{n \geqslant 0} z^n = (1-z)^{-1}
|
Generalized hypergeometric function |
|
|
- |
|
- | - | - | Empty arguments did not match the semantic macros (bug). | Full data:
{
"id": 20,
"pid": 70,
"eid": "math.70.58",
"title": "Generalized hypergeometric function",
"formulae": [
{
"id": "FORMULA_699b5f465d21dd6af7212cd8414f60c6",
"formula": "{}_1F_0(1;;z) = \\sum_{n \\geqslant 0} z^n = (1-z)^{-1}",
"semanticFormula": "\\genhyperF{1}{0}@{1}{}{z} = \\sum_{n \\geqslant 0} z^n = (1-z)^{-1}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "HypergeometricPFQ[{1}, {}, z] == Sum[(z)^(n), {n, 0, Infinity}, GenerateConditions->None] == (1 - z)^(- 1)"
},
"Maple": {
"translation": "hypergeom([1], [], z) = sum((z)^(n), n = 0..infinity) = (1 - z)^(- 1)"
}
},
"positions": [
{
"section": 17,
"sentence": 2,
"word": 0
}
],
"includes": [
"z",
"n",
"z)",
"_{p}F_{q}",
"^{n}"
],
"isPartOf": [],
"definiens": [
{
"definition": "geometric series with ratio",
"score": 2
},
{
"definition": "coefficient",
"score": 0
},
{
"definition": "hypergeometric function",
"score": 2
}
]
}
]
}
|
| Gold 21 | \chi(-1) = 1
|
Dirichlet L-function |
|
|
- | - | - |
|
- | It was translated to DirichletCharacter[1, k, - 1] == 1. The only valid input for k is 1.
|
Full data:
{
"id": 21,
"pid": 71,
"eid": "math.71.1-1",
"title": "Dirichlet L-function",
"formulae": [
{
"id": "FORMULA_dcb9beab8f504cfc907c3165d24e5ad3",
"formula": "\\chi(-1) = 1",
"semanticFormula": "\\Dirichletchar@@{- 1}{k} = 1",
"confidence": 0.746792096089683,
"translations": {
"Mathematica": {
"translation": "DirichletCharacter[1, 1, -1] == 1"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 7
}
],
"includes": [
"\\chi"
],
"isPartOf": [
"a=\\begin{cases}0;&\\mbox{if }\\chi(-1)=1, \\\\ 1;&\\mbox{if }\\chi(-1)=-1,\\end{cases}"
],
"definiens": [
{
"definition": "primitive character",
"score": 2
},
{
"definition": "integer",
"score": 1
},
{
"definition": "only zero",
"score": 0
},
{
"definition": "Gamma function",
"score": 0
},
{
"definition": "symbol",
"score": 0
},
{
"definition": "functional equation",
"score": 0
},
{
"definition": "Gauss sum",
"score": 0
},
{
"definition": "Dirichlet character",
"score": 2
}
]
}
]
}
|
| Gold 22 | \operatorname{Bi}'(z)\sim \frac{z^{\frac{1}{4}}e^{\frac{2}{3}z^{\frac{3}{2}}}}{\sqrt\pi\,}\left[ \sum_{n=0}^{\infty}\frac{1+6n}{1-6n} \dfrac{ \Gamma(n+\frac{5}{6})\Gamma(n+\frac{1}{6})\left(\frac{3}{4}\right)^n}{2\pi n! z^{3n/2}} \right]
|
Airy function |
|
- | - | - |
|
- | - | No translation possible for \sim
|
Full data:
{
"id": 22,
"pid": 72,
"eid": "math.72.15",
"title": "Airy function",
"formulae": [
{
"id": "FORMULA_3b2520d05d324290456841271e8d565b",
"formula": "\\operatorname{Bi}'(z)\\sim \\frac{z^{\\frac{1}{4}}e^{\\frac{2}{3}z^{\\frac{3}{2}}}}{\\sqrt\\pi\\,}\\left[ \\sum_{n=0}^{\\infty}\\frac{1+6n}{1-6n} \\dfrac{ \\Gamma(n+\\frac{5}{6})\\Gamma(n+\\frac{1}{6})\\left(\\frac{3}{4}\\right)^n}{2\\pi n! z^{3n\/2}} \\right]",
"semanticFormula": "\\AiryBi'@{z} \\sim \\frac{z^{\\frac{1}{4}} \\expe^{\\frac{2}{3}z^{\\frac{3}{2}}}}{\\sqrt{\\cpi}} [\\sum_{n=0}^{\\infty} \\frac{1+6n}{1-6n} \\frac{\\EulerGamma@{n + \\frac{5}{6}} \\EulerGamma@{n + \\frac{1}{6}}(\\frac{3}{4})^n{2 \\cpi n! z^{3n\/2}}}]",
"confidence": 0.6525418663370697,
"translations": {},
"positions": [
{
"section": 3,
"sentence": 9,
"word": 9
}
],
"includes": [
"z",
"z)",
"= 0"
],
"isPartOf": [],
"definiens": [
{
"definition": "z",
"score": 0
},
{
"definition": "asymptotic formula for Ai",
"score": 1
},
{
"definition": "Bi",
"score": 1
},
{
"definition": "asymptotic behaviour of the Airy function",
"score": 1
},
{
"definition": "Ai",
"score": 1
},
{
"definition": "cosine",
"score": 2
},
{
"definition": "definition of the Airy function",
"score": 1
},
{
"definition": "Airy function",
"score": 2
},
{
"definition": "Gamma function",
"score": 2
}
]
}
]
}
|
| Gold 23 | F'(y)=1-2yF(y)
|
Dawson function |
|
|
- | - |
|
|
- | No dependency to Dawson. | Full data:
{
"id": 23,
"pid": 73,
"eid": "math.73.41",
"title": "Dawson function",
"formulae": [
{
"id": "FORMULA_f6b555bd8ce626d90119ab5eafdaeff2",
"formula": "F'(y)=1-2yF(y)",
"semanticFormula": "\\DawsonsintF'@{y}=1-2y\\DawsonsintF@{y}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "D[DawsonF[y], {y, 1}] == 1 - 2*y*DawsonF[y]"
},
"Maple": {
"translation": "diff( dawson(y), y$(1) ) = 1 - 2*y*dawson(y)"
}
},
"positions": [
{
"section": 2,
"sentence": 9,
"word": 1
}
],
"includes": [
"y"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 0
},
{
"definition": "Dawson function",
"score": 2
}
]
}
]
}
|
| Gold 24 | s\not =1
|
Hurwitz zeta function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 24,
"pid": 74,
"eid": "math.74.0-1",
"title": "Hurwitz zeta function",
"formulae": [
{
"id": "FORMULA_80a3608d4c2aae63f082861007c16c38",
"formula": "s\\not =1",
"semanticFormula": "s \\neq 1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "s \\[NotEqual] 1"
}
},
"positions": [
{
"section": 0,
"sentence": 2,
"word": 24
}
],
"includes": [
"s",
"1",
"\\not = 1"
],
"isPartOf": [],
"definiens": [
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| Gold 25 | q = e^{i\pi\tau}
|
Theta function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 25,
"pid": 75,
"eid": "math.75.6-1",
"title": "Theta function",
"formulae": [
{
"id": "FORMULA_bfba6c35dbbcd8b89c6a29b1ffd6f517",
"formula": "q = e^{i\\pi\\tau}",
"semanticFormula": "q = \\expe^{\\iunit \\cpi \\tau}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "q == Exp[I*Pi*\\[Tau]]"
},
"Maple": {
"translation": "q = exp(I*Pi*tau)"
}
},
"positions": [
{
"section": 2,
"sentence": 0,
"word": 57
}
],
"includes": [
"\\tau",
"q",
"w = e^{\\pi iz}",
"q = e^{\\pi i\\tau}"
],
"isPartOf": [
"q = e^{\\pi i\\tau}",
"q = e^{2\\pi i\\tau}",
"\\theta_F (z)= \\sum_{m\\in \\Z^n} e^{2\\pi izF(m)}",
"\\hat{\\theta}_F (z) = \\sum_{k=0}^\\infty R_F(k) e^{2\\pi ikz}"
],
"definiens": [
{
"definition": "term of the nome",
"score": 2
},
{
"definition": "nome",
"score": 2
}
]
}
]
}
|
| Gold 26 | \frac{\mathrm{d}}{\mathrm{d}z} \operatorname{dn}(z) = - k^2 \operatorname{sn}(z) \operatorname{cn}(z)
|
Jacobi elliptic functions |
|
|
- | - | - | - | - | - | Full data:
{
"id": 26,
"pid": 76,
"eid": "math.76.155",
"title": "Jacobi elliptic functions",
"formulae": [
{
"id": "FORMULA_b54c03865b3efa9ea9112567cd66f59d",
"formula": "\\frac{\\mathrm{d}}{\\mathrm{d}z} \\operatorname{dn}(z) = - k^2 \\operatorname{sn}(z) \\operatorname{cn}(z)",
"semanticFormula": "\\deriv [1]{ }{z} \\Jacobielldnk@@{(z)}{k} = - k^2 \\Jacobiellsnk@@{(z)}{k} \\Jacobiellcnk@@{(z)}{k}",
"confidence": 0.6954186066124032,
"translations": {
"Mathematica": {
"translation": "D[JacobiDN[z, (k)^2], {z, 1}] == - (k)^(2)* JacobiSN[z, (k)^2]*JacobiCN[z, (k)^2]"
},
"Maple": {
"translation": "diff(JacobiDN(z, k), [z$(1)]) = - (k)^(2)* JacobiSN(z, k)*JacobiCN(z, k)"
}
},
"positions": [
{
"section": 16,
"sentence": 0,
"word": 15
}
],
"includes": [
"k",
"^{2}"
],
"isPartOf": [],
"definiens": [
{
"definition": "derivative",
"score": 2
},
{
"definition": "elliptic function",
"score": 2
},
{
"definition": "basic Jacobi",
"score": 0
},
{
"definition": "sn",
"score": 2
},
{
"definition": "dn",
"score": 2
},
{
"definition": "cn",
"score": 2
}
]
}
]
}
|
| Gold 27 | \int_{-\infty}^\infty \frac {\gamma\left(\frac s 2, z^2 \pi \right)} {(z^2 \pi)^\frac s 2} e^{-2 \pi i k z} \mathrm d z = \frac {\Gamma\left(\frac {1-s} 2, k^2 \pi \right)} {(k^2 \pi)^\frac {1-s} 2}
|
Incomplete gamma function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 27,
"pid": 77,
"eid": "math.77.118",
"title": "Incomplete gamma function",
"formulae": [
{
"id": "FORMULA_c82b4ceebacd2b4a03b2eff406834e61",
"formula": "\\int_{-\\infty}^\\infty \\frac {\\gamma\\left(\\frac s 2, z^2 \\pi \\right)} {(z^2 \\pi)^\\frac s 2} e^{-2 \\pi i k z} \\mathrm d z = \\frac {\\Gamma\\left(\\frac {1-s} 2, k^2 \\pi \\right)} {(k^2 \\pi)^\\frac {1-s} 2}",
"semanticFormula": "\\int_{-\\infty}^\\infty \\frac{\\incgamma@{\\frac s 2}{z^2 \\cpi}}{(z^2 \\cpi)^\\frac s 2} \\expe^{- 2 \\cpi \\iunit k z} \\diff{z} = \\frac{\\incGamma@{\\frac {1-s} 2}{k^2 \\cpi}}{(k^2 \\cpi)^\\frac {1-s} 2}}",
"confidence": 0.8121295595054496,
"translations": {
"Mathematica": {
"translation": "Integrate[Divide[Gamma[Divide[s,2], 0, (z)^(2)* Pi],((z)^(2)* Pi)^(Divide[s,2])]*Exp[- 2*Pi*I*k*z], {z, - Infinity, Infinity}, GenerateConditions->None] == Divide[Gamma[Divide[1 - s,2], (k)^(2)* Pi],((k)^(2)* Pi)^(Divide[1 - s,2])]"
},
"Maple": {
"translation": "int((GAMMA((s)\/(2))-GAMMA((s)\/(2), (z)^(2)* Pi))\/(((z)^(2)* Pi)^((s)\/(2)))*exp(- 2*Pi*I*k*z), z = - infinity..infinity) = (GAMMA((1 - s)\/(2), (k)^(2)* Pi))\/(((k)^(2)* Pi)^((1 - s)\/(2)))"
}
},
"positions": [
{
"section": 25,
"sentence": 1,
"word": 15
}
],
"includes": [
"\\gamma(s, z)",
"\\gamma",
"z^s",
"\\Gamma",
"\\gamma(s,z)",
"k",
"z",
"z=",
"2\\pi",
"\\gamma(u,v)",
"\\gamma(s,x)",
"s",
"z^{s}",
"e^{-x}",
"\\gamma(a,x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Fourier",
"score": 1
},
{
"definition": "upper incomplete Gamma function",
"score": 2
},
{
"definition": "lower incomplete Gamma function",
"score": 2
}
]
}
]
}
|
| Gold 28 | _{1}(z) =
|
Polylogarithm |
|
|
- | - | - | - | - | Wrong math detection. | Full data:
{
"id": 28,
"pid": 78,
"eid": "math.78.0-1",
"title": "Polylogarithm",
"formulae": [
{
"id": "FORMULA_e939f30d07578c2fb0d8cb5201db3c79",
"formula": "_{1}(z) =",
"semanticFormula": "\\polylog{1}@{z} = -\\ln@{1-z}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "PolyLog[1, z] = -Log[1 - z]"
},
"Maple": {
"translation": "polylog(1, z) = -ln(1 - z)"
}
},
"positions": [
{
"section": 0,
"sentence": 7,
"word": 11
}
],
"includes": [
"_{1}",
"z",
"z) =",
"z)",
"1"
],
"isPartOf": [
"\\operatorname{Li}_{1}(z) = -\\ln(1-z)",
"\\operatorname{Ti}_0(z) = {z \\over 1+z^2}, \\quad \\operatorname{Ti}_1(z) = \\arctan z, \\quad \\operatorname{Ti}_2(z) = \\int_0^z {\\arctan t \\over t} dt, \\quad \\ldots\\quad \\operatorname{Ti}_{n+1}(z) = \\int_0^z \\frac{\\operatorname{Ti}_n(t)}{t} dt"
],
"definiens": [
{
"definition": "natural logarithm",
"score": 2
},
{
"definition": "logarithm",
"score": 2
},
{
"definition": "polylogarithm function",
"score": 2
},
{
"definition": "dilogarithm",
"score": 1
},
{
"definition": "trilogarithm",
"score": 1
}
]
}
]
}
|
| Gold 29 | \int_{-\infty}^\infty \operatorname{sinc}(t) \, e^{-i 2 \pi f t}\,dt = \operatorname{rect}(f)
|
Sinc function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 29,
"pid": 79,
"eid": "math.79.11",
"title": "Sinc function",
"formulae": [
{
"id": "FORMULA_6340f4a043f912a3557e084aaf03792a",
"formula": "\\int_{-\\infty}^\\infty \\operatorname{sinc}(t) \\, e^{-i 2 \\pi f t}\\,dt = \\operatorname{rect}(f)",
"semanticFormula": "\\int_{-\\infty}^\\infty \\operatorname{sinc}(t) \\expe^{- \\iunit 2 \\cpi f t} \\diff{t} = \\operatorname{rect}(f)",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Integrate[sinc[(t)]*Exp[- I*2*Pi*f*t], {t, - Infinity, Infinity}, GenerateConditions->None] == rect[f]"
},
"Maple": {
"translation": "int(sinc((t))*exp(- I*2*Pi*f*t), t = - infinity..infinity) = rect(f)"
}
},
"positions": [
{
"section": 1,
"sentence": 9,
"word": 16
}
],
"includes": [
"\\pi",
"\\infty",
"sinc"
],
"isPartOf": [],
"definiens": [
{
"definition": "argument",
"score": 0
},
{
"definition": "continuous Fourier",
"score": 2
},
{
"definition": "rectangular function",
"score": 2
},
{
"definition": "sinc",
"score": 2
},
{
"definition": "ordinary frequency",
"score": 1
}
]
}
]
}
|
| Gold 30 | N=1
|
Exponential integral |
|
|
- | - | - | - | - | - | Full data:
{
"id": 30,
"pid": 80,
"eid": "math.80.26",
"title": "Exponential integral",
"formulae": [
{
"id": "FORMULA_a9a738ef9d4e46360dd9b87b39c691bf",
"formula": "N=1",
"semanticFormula": "N=1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "N == 1"
},
"Maple": {
"translation": "N = 1"
}
},
"positions": [
{
"section": 4,
"sentence": 3,
"word": 30
}
],
"includes": [
"N"
],
"isPartOf": [],
"definiens": [
{
"definition": "large value",
"score": 2
},
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| Gold 31 | \sum_{n=0}^\infty \frac{n!\,\Gamma\left(\alpha + 1\right)}{\Gamma\left(n+\alpha+1\right)}L_n^{(\alpha)}(x)L_n^{(\alpha)}(y)t^n=\frac{1}{(1-t)^{\alpha + 1}}e^{-(x+y)t/(1-t)}\,_0F_1\left(;\alpha + 1;\frac{xyt}{(1-t)^2}\right)
|
Laguerre polynomials |
|
|
- | - | - |
|
- | No infos about the gamma function. | Full data:
{
"id": 31,
"pid": 81,
"eid": "math.81.84",
"title": "Laguerre polynomials",
"formulae": [
{
"id": "FORMULA_f179a85d8102cbedb67cf60b188a68b7",
"formula": "\\sum_{n=0}^\\infty \\frac{n!\\,\\Gamma\\left(\\alpha + 1\\right)}{\\Gamma\\left(n+\\alpha+1\\right)}L_n^{(\\alpha)}(x)L_n^{(\\alpha)}(y)t^n=\\frac{1}{(1-t)^{\\alpha + 1}}e^{-(x+y)t\/(1-t)}\\,_0F_1\\left(;\\alpha + 1;\\frac{xyt}{(1-t)^2}\\right)",
"semanticFormula": "\\sum_{n=0}^\\infty \\frac{n! \\EulerGamma@{\\alpha + 1}}{\\EulerGamma@{n + \\alpha + 1}} \\LaguerrepolyL[\\alpha]{n}@{x} \\LaguerrepolyL[\\alpha]{n}@{x} t^n = \\frac{1}{(1-t)^{\\alpha + 1}} \\expe^{-(x+y)t\/(1-t)} \\genhyperF{0}{1}@{}{\\alpha + 1}{\\frac{xyt}{(1-t)^2}}",
"confidence": 0.8953028732079359,
"translations": {
"Mathematica": {
"translation": "Sum[Divide[(n)!*Gamma[\\[Alpha]+ 1],Gamma[n + \\[Alpha]+ 1]]*LaguerreL[n, \\[Alpha], x]*LaguerreL[n, \\[Alpha], x]*(t)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - t)^(\\[Alpha]+ 1)]*Exp[-(x + y)*t\/(1 - t)]*HypergeometricPFQ[{}, {\\[Alpha]+ 1}, Divide[x*y*t,(1 - t)^(2)]]"
},
"Maple": {
"translation": "sum((factorial(n)*GAMMA(alpha + 1))\/(GAMMA(n + alpha + 1))*LaguerreL(n, alpha, x)*LaguerreL(n, alpha, x)*(t)^(n), n = 0..infinity) = (1)\/((1 - t)^(alpha + 1))*exp(-(x + y)*t\/(1 - t))*hypergeom([], [alpha + 1], (x*y*t)\/((1 - t)^(2)))"
}
},
"positions": [
{
"section": 15,
"sentence": 0,
"word": 10
}
],
"includes": [
"\\alpha",
"L_{n}^{(\\alpha)}",
"L_n^{(\\alpha)}(x)",
"n"
],
"isPartOf": [],
"definiens": [
{
"definition": "Hille formula",
"score": 2
},
{
"definition": "Laguerre polynomial",
"score": 2
},
{
"definition": "series on the left converge",
"score": 0
},
{
"definition": "generalized Laguerre polynomial",
"score": 2
},
{
"definition": "confluent hypergeometric function",
"score": 2
}
]
}
]
}
|
| Gold 32 | c_{lm} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!}
|
Associated Legendre polynomials |
|
|
- | - | - | - | - | - | Full data:
{
"id": 32,
"pid": 82,
"eid": "math.82.8",
"title": "Associated Legendre polynomials",
"formulae": [
{
"id": "FORMULA_6f29e15c07089506a70db1b3f54b27a5",
"formula": "c_{lm} = (-1)^m \\frac{(\\ell-m)!}{(\\ell+m)!}",
"semanticFormula": "c_{lm} = (-1)^m \\frac{(\\ell-m)!}{(\\ell+m)!}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Subscript[c, l, m] == (- 1)^(m)*Divide[(\\[ScriptL]- m)!,(\\[ScriptL]+ m)!]"
},
"Maple": {
"translation": "c[l, m] = (- 1)^(m)*(factorial(ell - m))\/(factorial(ell + m))"
}
},
"positions": [
{
"section": 1,
"sentence": 7,
"word": 26
}
],
"includes": [
"m",
"(-1)^{m}",
"- 1"
],
"isPartOf": [],
"definiens": [
{
"definition": "proportionality constant",
"score": 2
}
]
}
]
}
|
| Gold 33 | \mathrm{Gi}(x) = \frac{1}{\pi} \int_0^\infty \sin\left(\frac{t^3}{3} + xt\right)\, dt
|
Scorer's function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 33,
"pid": 83,
"eid": "math.83.3",
"title": "Scorer's function",
"formulae": [
{
"id": "FORMULA_c8116180276232704ca3e9f67f207565",
"formula": "\\mathrm{Gi}(x) = \\frac{1}{\\pi} \\int_0^\\infty \\sin\\left(\\frac{t^3}{3} + xt\\right)\\, dt",
"semanticFormula": "\\ScorerGi@{x} = \\frac{1}{\\cpi} \\int_0^\\infty \\sin(\\frac{t^3}{3} + xt) \\diff{t}",
"confidence": 0.7929614010341081,
"translations": {
"Mathematica": {
"translation": "ScorerGi[x] == Divide[1,Pi]*Integrate[Sin[Divide[(t)^(3),3]+ x*t], {t, 0, Infinity}, GenerateConditions->None]",
"translationInformation": {
"subEquations": [
"ScorerGi[x] = Divide[1,Pi]*Integrate[Sin[Divide[(t)^(3),3]+ x*t], {t, 0, Infinity}, GenerateConditions->None]"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\ScorerGi": "Scorer function Gi; Example: \\ScorerGi@{z}\nWill be translated to: ScorerGi[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/9.12#i\nMathematica: https:\/\/",
"\\cpi": "Pi was translated to: Pi",
"\\sin": "Sine; Example: \\sin@@{z}\nWill be translated to: Sin[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E1\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Sin.html"
}
}
},
"Maple": {
"translation": "AiryBi(x)*(int(AiryAi(t), t = (x) .. infinity))+AiryAi(x)*(int(AiryBi(t), t = 0 .. (x))) = (1)\/(Pi)*int(sin(((t)^(3))\/(3)+ x*t), t = 0..infinity)",
"translationInformation": {
"subEquations": [
"AiryBi(x)*(int(AiryAi(t), t = (x) .. infinity))+AiryAi(x)*(int(AiryBi(t), t = 0 .. (x))) = (1)\/(Pi)*int(sin(((t)^(3))\/(3)+ x*t), t = 0..infinity)"
],
"freeVariables": [
"x"
],
"constraints": [],
"tokenTranslations": {
"\\ScorerGi": "Scorer function Gi; Example: \\ScorerGi@{z}\nWill be translated to: \nAlternative translations: [AiryBi($0)*(int(AiryAi(t), t = ($0) .. infinity))+AiryAi($0)*(int(AiryBi(t), t = 0 .. ($0)))]Relevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/9.12#i\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=Airy",
"\\cpi": "Pi was translated to: Pi",
"\\sin": "Sine; Example: \\sin@@{z}\nWill be translated to: sin($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/4.14#E1\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=sin"
}
}
}
},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 15
}
],
"includes": [
"x",
"x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Scorer 's function",
"score": 2
},
{
"definition": "special function",
"score": 1
}
]
}
]
}
|
| Gold 34 | \frac{\partial^2}{\partial x^2} V(x;\sigma,\gamma)= \frac{x^2-\gamma^2-\sigma^2}{\sigma^4} \frac{\operatorname{Re}[w(z)]}{\sigma\sqrt{2 \pi}}-\frac{2 x \gamma}{\sigma^4} \frac{\operatorname{Im}[w(z)]}{\sigma\sqrt{2 \pi}}+\frac{\gamma}{\sigma^4}\frac{1}{\pi}
|
Voigt profile |
|
|
- | - | - | - |
|
- | Full data:
{
"id": 34,
"pid": 84,
"eid": "math.84.31",
"title": "Voigt profile",
"formulae": [
{
"id": "FORMULA_e663d20df3cca1ae5dec645d320cd511",
"formula": "\\frac{\\partial^2}{\\partial x^2} V(x;\\sigma,\\gamma)= \\frac{x^2-\\gamma^2-\\sigma^2}{\\sigma^4} \\frac{\\operatorname{Re}[w(z)]}{\\sigma\\sqrt{2 \\pi}}-\\frac{2 x \\gamma}{\\sigma^4} \\frac{\\operatorname{Im}[w(z)]}{\\sigma\\sqrt{2 \\pi}}+\\frac{\\gamma}{\\sigma^4}\\frac{1}{\\pi}",
"semanticFormula": "\\deriv[2]{}{x} V(x ; \\sigma , \\gamma) = \\frac{x^2-\\gamma^2-\\sigma^2}{\\sigma^4} \\frac{\\realpart [\\Faddeevaw@{z}]}{\\sigma \\sqrt{2 \\cpi}} - \\frac{2 x \\gamma}{\\sigma^4} \\frac{\\imagpart [\\Faddeevaw@{z}]}{\\sigma \\sqrt{2 \\cpi}} + \\frac{\\gamma}{\\sigma^4} \\frac{1}{\\cpi}",
"confidence": 0.8620216359266987,
"translations": {
"Mathematica": {
"translation": "D[PDF[VoigtDistribution[\\[Gamma], \\[Sigma]], x], {x, 2}] == Divide[x^2 - \\[Gamma]^2 - \\[Sigma]^2, \\[Sigma]^4] * Divide[ Re[ Exp[-(Divide[x+I*y,\\[Sigma]*Sqrt[2]])^2]*Erfc[-I*(Divide[x+I*y,\\[Sigma]*Sqrt[2]])] ], \\[Sigma]*Sqrt[2*Pi]] - Divide[2*x*y, \\[Sigma]^4] * Divide[Im[Exp[-(Divide[x+I*y,\\[Sigma]*Sqrt[2]])^2]*Erfc[-I*(Divide[x+I*y,\\[Sigma]*Sqrt[2]])]], \\[Sigma]*Sqrt[2*Pi]] + Divide[\\[Gamma],\\[Sigma]^4]*Divide[1,Pi]"
}
},
"positions": [
{
"section": 6,
"sentence": 0,
"word": 20
}
],
"includes": [
"w(z)]",
"z",
"V(x;\\sigma,\\gamma)",
"x",
"w(z)"
],
"isPartOf": [],
"definiens": [
{
"definition": "term of the Faddeeva function",
"score": 2
},
{
"definition": "second derivative profile",
"score": 2
},
{
"definition": "real part of the Faddeeva function",
"score": 2
},
{
"definition": "Faddeeva function",
"score": 2
},
{
"definition": "Voigt function",
"score": 2
},
{
"definition": "Voigt profile",
"score": 2
}
]
}
]
}
|
| Gold 35 | \Phi(z,s,a) = \frac{1}{1-z} \frac{1}{a^{s}} + \sum_{n=1}^{N-1} \frac{(-1)^{n} \mathrm{Li}_{-n}(z)}{n!} \frac{(s)_{n}}{a^{n+s}} +O(a^{-N-s})
|
Lerch zeta function |
|
|
- | - | - | - |
|
Landau notation. | Full data:
{
"id": 35,
"pid": 85,
"eid": "math.85.57",
"title": "Lerch zeta function",
"formulae": [
{
"id": "FORMULA_a0cc62efe3cabac6d8bebe5b8b94b5fa",
"formula": "\\Phi(z,s,a) = \\frac{1}{1-z} \\frac{1}{a^{s}} + \\sum_{n=1}^{N-1} \\frac{(-1)^{n} \\mathrm{Li}_{-n}(z)}{n!} \\frac{(s)_{n}}{a^{n+s}} +O(a^{-N-s})",
"semanticFormula": "\\Phi(z , s , a) = \\frac{1}{1-z} \\frac{1}{a^{s}} + \\sum_{n=1}^{N-1} \\frac{(-1)^{n} \\polylog{-n}@{z}}{n!} \\frac{\\Pochhammersym{s}{n}}{a^{n+s}} + \\bigO{a^{-N-s}}",
"confidence": 0.8662724998444776,
"translations": {
"Mathematica": {
"translation": "\\[CapitalPhi][z, s, a] == Divide[1,1 - z]*Divide[1,(a)^(s)]+ Sum[Divide[(- 1)^(n)* PolyLog[-n, z],(n)!]*Divide[Pochhammer[s, n],(a)^(n + s)], {n, 1, N - 1}, GenerateConditions->None]+ O[a]^(- N - s)"
}
},
"positions": [
{
"section": 6,
"sentence": 1,
"word": 23
}
],
"includes": [
"a",
"\\Phi(z,s,a)",
"z",
"s"
],
"isPartOf": [],
"definiens": [
{
"definition": "asymptotic expansion",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "Lerch transcendent",
"score": 2
},
{
"definition": "polylogarithm",
"score": 2
},
{
"definition": "polylogarithm function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 2
}
]
}
]
}
|
| Gold 36 | M(1,2,z)=(e^z-1)/z,\ \ M(1,3,z)=2!(e^z-1-z)/z^2
|
Confluent hypergeometric function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 36,
"pid": 86,
"eid": "math.86.44",
"title": "Confluent hypergeometric function",
"formulae": [
{
"id": "FORMULA_d83a3ce5244b566d8f71edb7f81afa43",
"formula": "M(1,2,z)=(e^z-1)\/z,\\ \\ M(1,3,z)=2!(e^z-1-z)\/z^2",
"semanticFormula": "\\KummerconfhyperM@{1}{2}{z} = (\\expe^z - 1) \/ z , \\KummerconfhyperM@{1}{3}{z} = 2! (\\expe^z - 1 - z) \/ z^2",
"confidence": 0.912945064646862,
"translations": {
"Mathematica": {
"translation": "Hypergeometric1F1[1, 2, z] == (Exp[z]- 1)\/z\n Hypergeometric1F1[1, 3, z] == (2)!*(Exp[z]- 1 - z)\/(z)^(2)"
},
"Maple": {
"translation": "KummerM(1, 2, z) = (exp(z)- 1)\/z; KummerM(1, 3, z) = factorial(2)*(exp(z)- 1 - z)\/(z)^(2)"
}
},
"positions": [
{
"section": 10,
"sentence": 4,
"word": 0
}
],
"includes": [
"M",
"U(a, b, z)",
"z",
"U(n,c,z)",
"\\Phi(a, b, z)",
"M(n,b,z)",
"M(a, b, z)"
],
"isPartOf": [],
"definiens": [
{
"definition": "etc",
"score": 0
},
{
"definition": "Kummer 's function of the first kind",
"score": 2
},
{
"definition": "confluent hypergeometric function",
"score": 1
},
{
"definition": "hypergeometric function",
"score": 1
}
]
}
]
}
|
| Gold 37 | \sigma = \pm 1
|
Mathieu function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 37,
"pid": 87,
"eid": "math.87.54",
"title": "Mathieu function",
"formulae": [
{
"id": "FORMULA_f694135eafc20195a9d96ca3ce8af674",
"formula": "\\sigma = \\pm 1",
"semanticFormula": "\\sigma = \\pm 1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "\\[Sigma] == \\[PlusMinus]1",
"translationInformation": {
"subEquations": [
"\\[Sigma] = + 1",
"\\[Sigma] = - 1"
],
"freeVariables": [
"\\[Sigma]"
],
"constraints": [],
"tokenTranslations": {
"\\pm": "was translated to: \\[PlusMinus]"
}
}
},
"Maple": {
"translation": "sigma = &+- 1",
"translationInformation": {
"subEquations": [
"sigma = + 1",
"sigma = - 1"
],
"freeVariables": [
"sigma"
],
"constraints": [],
"tokenTranslations": {
"\\pm": "was translated to: &+-"
}
}
}
},
"positions": [
{
"section": 4,
"sentence": 1,
"word": 27
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| Gold 38 | \frac{d^2f}{dz^2} + \left(\tilde{a}z^2+\tilde{b}z+\tilde{c}\right)f=0
|
Parabolic cylinder function |
|
|
- | - | - |
|
- | ODE. f does not show the argument z. | Full data:
{
"id": 38,
"pid": 88,
"eid": "math.88.0",
"title": "Parabolic cylinder function",
"formulae": [
{
"id": "FORMULA_bec6388631b20f2af14e375b13e1533f",
"formula": "\\frac{d^2f}{dz^2} + \\left(\\tilde{a}z^2+\\tilde{b}z+\\tilde{c}\\right)f=0",
"semanticFormula": "\\deriv [2]{f}{z} +(\\tilde{a} z^2 + \\tilde{b} z + \\tilde{c}) f = 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "D[f[z], {z, 2}] + (a*z^2 + b*z + c)*f[z] == 0"
}
},
"positions": [
{
"section": 0,
"sentence": 0,
"word": 19
}
],
"includes": [
"z"
],
"isPartOf": [],
"definiens": [
{
"definition": "solution to the differential equation",
"score": 2
},
{
"definition": "special function",
"score": 1
},
{
"definition": "mathematics",
"score": 0
},
{
"definition": "parabolic cylinder function",
"score": 1
}
]
}
]
}
|
| Gold 39 | c=\infty
|
Painlevé transcendents |
|
|
- | - | - | - | - | - | Full data:
{
"id": 39,
"pid": 89,
"eid": "math.89.23",
"title": "Painlev\u00e9 transcendents",
"formulae": [
{
"id": "FORMULA_0a306ab913684a1ba3935715d3dd8ad8",
"formula": "c=\\infty",
"semanticFormula": "c=\\infty",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "c == Infinity"
},
"Maple": {
"translation": "c = infinity"
}
},
"positions": [
{
"section": 9,
"sentence": 5,
"word": 23
}
],
"includes": [
"c"
],
"isPartOf": [],
"definiens": [
{
"definition": "central charge of the Virasoro algebra",
"score": 2
},
{
"definition": "combination of conformal block",
"score": 1
},
{
"definition": "Painlev\u00e9 VI equation",
"score": 1
},
{
"definition": "two-dimensional conformal field theory",
"score": 1
}
]
}
]
}
|
| Gold 40 | c = a + 1
|
Hypergeometric function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 40,
"pid": 90,
"eid": "math.90.7",
"title": "Hypergeometric function",
"formulae": [
{
"id": "FORMULA_aaffb0ad8dea17d68491d9fb6ebcfbe3",
"formula": "c = a + 1",
"semanticFormula": "c = a + 1",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "c == a + 1"
},
"Maple": {
"translation": "c := a + 1"
}
},
"positions": [
{
"section": 3,
"sentence": 0,
"word": 20
}
],
"includes": [
"a",
"c"
],
"isPartOf": [],
"definiens": [
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| Gold 41 | \frac{1}{\Gamma(z)}= z e^{\gamma z} \prod_{k=1}^\infty \left\{ \left(1+\frac{z}{k}\right)e^{-z/k} \right\}
|
Barnes G-function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 41,
"pid": 91,
"eid": "math.91.47",
"title": "Barnes G-function",
"formulae": [
{
"id": "FORMULA_6bc0d742c4d25c1abb61158150489676",
"formula": "\\frac{1}{\\Gamma(z)}= z e^{\\gamma z} \\prod_{k=1}^\\infty \\left\\{ \\left(1+\\frac{z}{k}\\right)e^{-z\/k} \\right\\}",
"semanticFormula": "\\frac{1}{\\EulerGamma@{z}} = z \\expe^{\\EulerConstant z} \\prod_{k=1}^\\infty \\{(1 + \\frac{z}{k}) \\expe^{-z\/k} \\}",
"confidence": 0.8614665289982916,
"translations": {
"Mathematica": {
"translation": "Divide[1,Gamma[z]] == z*Exp[EulerGamma*z]*Product[(1 +Divide[z,k])*Exp[- z\/k], {k, 1, Infinity}, GenerateConditions->None]",
"translationInformation": {
"subEquations": [
"Divide[1,Gamma[z]] = z*Exp[EulerGamma*z]*Product[(1 +Divide[z,k])*Exp[- z\/k], {k, 1, Infinity}, GenerateConditions->None]"
],
"freeVariables": [
"z"
],
"constraints": [],
"tokenTranslations": {
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\EulerConstant": "Euler-Mascheroni constant was translated to: EulerGamma",
"\\EulerGamma": "Euler Gamma function; Example: \\EulerGamma@{z}\nWill be translated to: Gamma[$0]\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/5.2#E1\nMathematica: https:\/\/reference.wolfram.com\/language\/ref\/Gamma.html"
}
}
},
"Maple": {
"translation": "(1)\/(GAMMA(z)) = z*exp(gamma*z)*product((1 +(z)\/(k))*exp(- z\/k), k = 1..infinity)",
"translationInformation": {
"subEquations": [
"(1)\/(GAMMA(z)) = z*exp(gamma*z)*product((1 +(z)\/(k))*exp(- z\/k), k = 1..infinity)"
],
"freeVariables": [
"z"
],
"constraints": [],
"tokenTranslations": {
"\\expe": "Recognizes e with power as the exponential function. It was translated as a function.",
"\\EulerConstant": "Euler-Mascheroni constant was translated to: gamma",
"\\EulerGamma": "Euler Gamma function; Example: \\EulerGamma@{z}\nWill be translated to: GAMMA($0)\nRelevant links to definitions:\nDLMF: http:\/\/dlmf.nist.gov\/5.2#E1\nMaple: https:\/\/www.maplesoft.com\/support\/help\/maple\/view.aspx?path=GAMMA"
}
}
}
},
"positions": [
{
"section": 8,
"sentence": 0,
"word": 55
}
],
"includes": [
"\\,\\Gamma(x)",
"\\, \\gamma",
"z",
"\\,\\gamma"
],
"isPartOf": [],
"definiens": [
{
"definition": "Euler",
"score": 1
},
{
"definition": "Mascheroni",
"score": 1
},
{
"definition": "gamma function",
"score": 2
}
]
}
]
}
|
| Gold 42 | 192/24 = 8 = 2 \times 4
|
Heun function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 42,
"pid": 92,
"eid": "math.92.1-1",
"title": "Heun function",
"formulae": [
{
"id": "FORMULA_8c78ef87048e61947a6d7d4b5e06aa63",
"formula": "192\/24 = 8 = 2 \\times 4",
"semanticFormula": "192\/24 = 8 = 2 \\times 4",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "192\/24 == 8 == 2 * 4",
"translationInformation": {
"subEquations": [
"192\/24 = 8",
"8 = 2 * 4"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\times": "was translated to: *"
}
},
"numericResults": {
"overallResult": "SUCCESS",
"numberOfTests": 2,
"numberOfFailedTests": 0,
"numberOfSuccessfulTests": 2,
"numberOfSkippedTests": 0,
"numberOfErrorTests": 0,
"wasAborted": false,
"crashed": false,
"testCalculationsGroups": [
{
"lhs": "192\/24",
"rhs": "8",
"testExpression": "(192\/24)-(8)",
"activeConstraints": [],
"testCalculations": [
{
"result": "SUCCESS",
"resultExpression": "0.",
"testValues": {}
}
]
},
{
"lhs": "8",
"rhs": "2 * 4",
"testExpression": "(8)-(2 * 4)",
"activeConstraints": [],
"testCalculations": [
{
"result": "SUCCESS",
"resultExpression": "0.",
"testValues": {}
}
]
}
]
},
"symbolicResults": {
"overallResult": "SUCCESS",
"numberOfTests": 2,
"numberOfFailedTests": 0,
"numberOfSuccessfulTests": 2,
"numberOfSkippedTests": 0,
"numberOfErrorTests": 0,
"crashed": false,
"testCalculationsGroup": [
{
"lhs": "192\/24",
"rhs": "8",
"testExpression": "(192\/24)-(8)",
"testCalculations": [
{
"result": "SUCCESS",
"testTitle": "Simple",
"testExpression": "FullSimplify[(192\/24)-(8)]",
"resultExpression": "0",
"wasAborted": false,
"conditionallySuccessful": false
}
]
},
{
"lhs": "8",
"rhs": "2 * 4",
"testExpression": "(8)-(2 * 4)",
"testCalculations": [
{
"result": "SUCCESS",
"testTitle": "Simple",
"testExpression": "FullSimplify[(8)-(2 * 4)]",
"resultExpression": "0",
"wasAborted": false,
"conditionallySuccessful": false
}
]
}
]
}
},
"SymPy": {
"translation": "192\/24 == 8 == 2 * 4",
"translationInformation": {
"subEquations": [
"192\/24 = 8",
"8 = 2 * 4"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\times": "was translated to: *"
}
}
},
"Maple": {
"translation": "192\/24 = 8 = 2 * 4",
"translationInformation": {
"subEquations": [
"192\/24 = 8",
"8 = 2 * 4"
],
"freeVariables": [],
"constraints": [],
"tokenTranslations": {
"\\times": "was translated to: *"
}
},
"numericResults": {
"overallResult": "SUCCESS",
"numberOfTests": 2,
"numberOfFailedTests": 0,
"numberOfSuccessfulTests": 2,
"numberOfSkippedTests": 0,
"numberOfErrorTests": 0,
"wasAborted": false,
"crashed": false,
"testCalculationsGroups": [
{
"lhs": "192\/24",
"rhs": "8",
"testExpression": "evalf((192\/24)-(8))",
"activeConstraints": [],
"testCalculations": [
{
"result": "SUCCESS",
"resultExpression": "0.",
"testValues": {}
}
]
},
{
"lhs": "8",
"rhs": "2 * 4",
"testExpression": "evalf((8)-(2 * 4))",
"activeConstraints": [],
"testCalculations": [
{
"result": "SUCCESS",
"resultExpression": "0.",
"testValues": {}
}
]
}
]
},
"symbolicResults": {
"overallResult": "SUCCESS",
"numberOfTests": 2,
"numberOfFailedTests": 0,
"numberOfSuccessfulTests": 2,
"numberOfSkippedTests": 0,
"numberOfErrorTests": 0,
"crashed": false,
"testCalculationsGroup": [
{
"lhs": "192\/24",
"rhs": "8",
"testExpression": "(192\/24)-(8)",
"testCalculations": [
{
"result": "SUCCESS",
"testTitle": "Simple",
"testExpression": "simplify((192\/24)-(8))",
"resultExpression": "0",
"wasAborted": false,
"conditionallySuccessful": false
}
]
},
{
"lhs": "8",
"rhs": "2 * 4",
"testExpression": "(8)-(2 * 4)",
"testCalculations": [
{
"result": "SUCCESS",
"testTitle": "Simple",
"testExpression": "simplify((8)-(2 * 4))",
"resultExpression": "0",
"wasAborted": false,
"conditionallySuccessful": false
}
]
}
]
}
}
},
"positions": [
{
"section": 3,
"sentence": 1,
"word": 25
}
],
"includes": [],
"isPartOf": [],
"definiens": []
}
]
}
|
| Gold 43 | =2
|
Gegenbauer polynomials |
|
|
- | - | - | - | - | Wrong math detection. | Full data:
{
"id": 43,
"pid": 93,
"eid": "math.93.0-1",
"title": "Gegenbauer polynomials",
"formulae": [
{
"id": "FORMULA_34d9d355f0c0e28d91465c3b575fb0a1",
"formula": "=2",
"semanticFormula": "\\alpha = 2",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "\\[Alpha] = 2"
},
"Maple": {
"translation": "alpha = 2"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 17
}
],
"includes": [],
"isPartOf": [
"\\begin{align}C_0^\\alpha(x) & = 1 \\\\C_1^\\alpha(x) & = 2 \\alpha x \\\\C_n^\\alpha(x) & = \\frac{1}{n}[2x(n+\\alpha-1)C_{n-1}^\\alpha(x) - (n+2\\alpha-2)C_{n-2}^\\alpha(x)].\\end{align}"
],
"definiens": [
{
"definition": "value",
"score": 2
}
]
}
]
}
|
| Gold 44 | \lim_{q\to 1}\;_{j}\phi_k \left[\begin{matrix} q^{a_1} & q^{a_2} & \ldots & q^{a_j} \\ q^{b_1} & q^{b_2} & \ldots & q^{b_k} \end{matrix} ; q,(q-1)^{1+k-j} z \right]=\;_{j}F_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_j \\ b_1 & b_2 & \ldots & b_k \end{matrix} ;z \right]
|
Basic hypergeometric series |
|
- | - |
|
- | - | - | Indef length of arguments are not translatable. | Full data:
{
"id": 44,
"pid": 94,
"eid": "math.94.4",
"title": "Basic hypergeometric series",
"formulae": [
{
"id": "FORMULA_33e3b57bb75d5ea3b5b8ddcceef38430",
"formula": "\\lim_{q\\to 1}\\;_{j}\\phi_k \\left[\\begin{matrix} q^{a_1} & q^{a_2} & \\ldots & q^{a_j} \\\\ q^{b_1} & q^{b_2} & \\ldots & q^{b_k} \\end{matrix} ; q,(q-1)^{1+k-j} z \\right]=\\;_{j}F_k \\left[\\begin{matrix} a_1 & a_2 & \\ldots & a_j \\\\ b_1 & b_2 & \\ldots & b_k \\end{matrix} ;z \\right]",
"semanticFormula": "\\lim_{q\\to 1} \\qgenhyperphi{j}{k}@{q^{a_1} , q^{a_2} , \\ldots , q^{a_j}}{q^{b_1} , q^{b_2} , \\ldots , q^{b_k}}{q}{(q - 1)^{1+k-j} z} = \\genhyperF{j}{k}@{a_1 , a_2 , \\ldots , a_j}{b_1 , b_2 , \\ldots , b_k}{z}",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 5,
"word": 13
}
],
"includes": [
"q^{n}",
"q",
"b",
"a",
"z"
],
"isPartOf": [],
"definiens": [
{
"definition": "q-analog of the hypergeometric series",
"score": 2
},
{
"definition": "unilateral basic hypergeometric series",
"score": 2
},
{
"definition": "basic hypergeometric series",
"score": 2
}
]
}
]
}
|
| Gold 45 | \frac{d^2w}{dz^2}+\left(-\frac{1}{4}+\frac{\kappa}{z}+\frac{1/4-\mu^2}{z^2}\right)w=0
|
Whittaker function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 45,
"pid": 95,
"eid": "math.95.0",
"title": "Whittaker function",
"formulae": [
{
"id": "FORMULA_16ec3a3583ee2b4621d316bf839c1725",
"formula": "\\frac{d^2w}{dz^2}+\\left(-\\frac{1}{4}+\\frac{\\kappa}{z}+\\frac{1\/4-\\mu^2}{z^2}\\right)w=0",
"semanticFormula": "\\deriv [2]{w}{z} +(- \\frac{1}{4} + \\frac{\\kappa}{z} + \\frac{1\/4-\\mu^2}{z^2}) w = 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "D[w, {z, 2}]+(-Divide[1,4]+Divide[\\[Kappa],z]+Divide[1\/4 - \\[Mu]^(2),(z)^(2)])*w == 0"
},
"Maple": {
"translation": "diff(w, [z$(2)])+(-(1)\/(4)+(kappa)\/(z)+(1\/4 - (mu)^(2))\/((z)^(2)))*w = 0"
}
},
"positions": [
{
"section": 0,
"sentence": 2,
"word": 4
}
],
"includes": [
"\\mu",
"\\kappa",
"z"
],
"isPartOf": [],
"definiens": [
{
"definition": "Whittaker 's equation",
"score": 2
},
{
"definition": "Whittaker function",
"score": 1
}
]
}
]
}
|
| Gold 46 | e_1=\tfrac12,\qquad e_2=0,\qquad e_3=-\tfrac12
|
Lemniscatic elliptic function |
|
|
- | - | - | - | - | Multi-equation problem (bug). | Full data:
{
"id": 46,
"pid": 96,
"eid": "math.96.1",
"title": "Lemniscatic elliptic function",
"formulae": [
{
"id": "FORMULA_24137d79f0a282f42fdf9ea93576e998",
"formula": "e_1=\\tfrac12,\\qquad e_2=0,\\qquad e_3=-\\tfrac12",
"semanticFormula": "e_1=\\tfrac12,\\qquad e_2=0,\\qquad e_3=-\\tfrac12",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Subscript[e, 1] == Divide[1,2]\n Subscript[e, 2] = 0\n Subscript[e, 3] = -Divide[1,2]"
},
"Maple": {
"translation": "e[1] := (1)\/(2); e[2] := 0; e[3] := -(1)\/(2)"
}
},
"positions": [
{
"section": 0,
"sentence": 5,
"word": 11
}
],
"includes": [
"e_{1}",
"e_{2}",
"e_{3}"
],
"isPartOf": [],
"definiens": [
{
"definition": "constant",
"score": 2
}
]
}
]
}
|
| Gold 47 | \gamma> 0,n-p=m-q> 0
|
Meijer G-function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 47,
"pid": 98,
"eid": "math.98.53-1",
"title": "Meijer G-function",
"formulae": [
{
"id": "FORMULA_028eb01ef675c90ea0f74fcdd93fc78c",
"formula": "\\gamma> 0,n-p=m-q> 0",
"semanticFormula": "\\gamma> 0,n-p=m-q> 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "\\[Gamma] > 0\n n - p == m - q > 0"
},
"Maple": {
"translation": "gamma > 0; n - p = m - q > 0"
}
},
"positions": [
{
"section": 12,
"sentence": 0,
"word": 17
}
],
"includes": [
"m",
"q",
"p=q> 0",
"n",
"p=q",
"\\gamma>"
],
"isPartOf": [],
"definiens": [
{
"definition": "constraint",
"score": 2
}
]
}
]
}
|
| Gold 48 | \begin{pmatrix} j \\ m \quad m'\end{pmatrix}:= \sqrt{2 j + 1}\begin{pmatrix} j & 0 & j \\ m & 0 & m'\end{pmatrix}= (-1)^{j - m'} \delta_{m, -m'}
|
3-j symbol |
|
|
- | - | - | - | - | LCT does not support matrix translations yet. | Full data:
{
"id": 48,
"pid": 99,
"eid": "math.99.30",
"title": "3-j symbol",
"formulae": [
{
"id": "FORMULA_3f987b881a59a03904ff9a79476faae0",
"formula": "\\begin{pmatrix} j \\\\ m \\quad m'\\end{pmatrix}:= \\sqrt{2 j + 1}\\begin{pmatrix} j & 0 & j \\\\ m & 0 & m'\\end{pmatrix}= (-1)^{j - m'} \\delta_{m, -m'}",
"semanticFormula": "\\begin{pmatrix} j \\\\ m \\quad m'\\end{pmatrix}:= \\sqrt{2 j + 1}\\begin{pmatrix} j & 0 & j \\\\ m & 0 & m'\\end{pmatrix}= (-1)^{j - m'} \\delta_{m, -m'}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Wigner[j_, m_, m\\[Prime]_] := Sqrt[2*j+1] * {{j, 0, j}, {m, 0, m\\[Prime]}} = (-1)^(j-m\\[Prime])*Subscript[\\[Delta], m, -m\\[Prime]]"
}
},
"positions": [
{
"section": 10,
"sentence": 0,
"word": 23
}
],
"includes": [
"j",
"m"
],
"isPartOf": [],
"definiens": [
{
"definition": "Wigner 1-jm symbol",
"score": 2
},
{
"definition": "metric tensor in angular-momentum theory",
"score": 2
},
{
"definition": "quantity",
"score": 0
}
]
}
]
}
|
| Gold 49 | \begin{Bmatrix} i & j & \ell\\ k & m & n \end{Bmatrix}= (\Phi_{i,j}^{k,m})_{\ell,n}
|
6-j symbol |
|
- | - | - | - |
|
- | A matrix cannot be defined as a function in Mathematica. | Full data:
{
"id": 49,
"pid": 100,
"eid": "math.100.14",
"title": "6-j symbol",
"formulae": [
{
"id": "FORMULA_21d6ec52b25bb130bf068c4857bbcb93",
"formula": "\\begin{Bmatrix} i & j & \\ell\\\\ k & m & n \\end{Bmatrix}= (\\Phi_{i,j}^{k,m})_{\\ell,n}",
"semanticFormula": "\\Wignersixjsym{i}{j}{\\ell}{k}{m}{n} = (\\Phi_{i,j}^{k,m})_{\\ell,n}",
"confidence": 0.8624533614429312,
"translations": {},
"positions": [
{
"section": 5,
"sentence": 4,
"word": 23
}
],
"includes": [
"j"
],
"isPartOf": [],
"definiens": [
{
"definition": "6j symbol",
"score": 2
},
{
"definition": "associativity isomorphism",
"score": 2
},
{
"definition": "symbol",
"score": 1
},
{
"definition": "vector space isomorphism",
"score": 2
},
{
"definition": "Wigner",
"score": 1
},
{
"definition": "Wigner 's 6 - j symbol",
"score": 2
}
]
}
]
}
|
| Gold 50 | \sum_{j_7 j_8} (2j_7+1)(2j_8+1) \begin{Bmatrix} j_1 & j_2 & j_3\\ j_4 & j_5 & j_6\\ j_7 & j_8 & j_9 \end{Bmatrix} \begin{Bmatrix} j_1 & j_2 & j_3'\\ j_4 & j_5 & j_6'\\ j_7 & j_8 & j_9 \end{Bmatrix} = \frac{\delta_{j_3j_3'}\delta_{j_6j_6'} \begin{Bmatrix} j_1 & j_2 & j_3 \end{Bmatrix} \begin{Bmatrix} j_4 & j_5 & j_6\end{Bmatrix} \begin{Bmatrix} j_3 & j_6 & j_9 \end{Bmatrix}} {(2j_3+1)(2j_6+1)}
|
9-j symbol |
|
- | - | - | - | - | - | Mistakenly interpreted as Wigner 6-j rather than 9-j. | Full data:
{
"id": 50,
"pid": 101,
"eid": "math.101.32",
"title": "9-j symbol",
"formulae": [
{
"id": "FORMULA_08d08037d9e64d85aa3645470ce645af",
"formula": "\\sum_{j_7 j_8} (2j_7+1)(2j_8+1) \\begin{Bmatrix} j_1 & j_2 & j_3\\\\ j_4 & j_5 & j_6\\\\ j_7 & j_8 & j_9 \\end{Bmatrix} \\begin{Bmatrix} j_1 & j_2 & j_3'\\\\ j_4 & j_5 & j_6'\\\\ j_7 & j_8 & j_9 \\end{Bmatrix} = \\frac{\\delta_{j_3j_3'}\\delta_{j_6j_6'} \\begin{Bmatrix} j_1 & j_2 & j_3 \\end{Bmatrix} \\begin{Bmatrix} j_4 & j_5 & j_6\\end{Bmatrix} \\begin{Bmatrix} j_3 & j_6 & j_9 \\end{Bmatrix}} {(2j_3+1)(2j_6+1)}",
"semanticFormula": "\\sum_{j_7 j_8} (2j_7+1)(2j_8+1) \\Wignerninejsym{j_1}{j_2}{j_3}{j_4}{j_5}{j_6}{j_7}{j_8}{j_9} \\Wignerninejsym{j_1}{j_2}{j_3'}{j_4}{j_5}{j_6'}{j_7}{j_8}{j_9} = \\frac{\\delta_{j_3j_3'}\\delta_{j_6j_6'} \\begin{Bmatrix} j_1 & j_2 & j_3 \\end{Bmatrix} \\begin{Bmatrix} j_4 & j_5 & j_6\\end{Bmatrix} \\begin{Bmatrix} j_3 & j_6 & j_9 \\end{Bmatrix}}{(2j_3+1)(2j_6+1)}",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 5,
"sentence": 0,
"word": 10
}
],
"includes": [
"j",
"_{4}"
],
"isPartOf": [],
"definiens": [
{
"definition": "orthogonality relation",
"score": 1
},
{
"definition": "triangular delta",
"score": 2
},
{
"definition": "symbol",
"score": 1
},
{
"definition": "Wigner 's 9 - j symbol",
"score": 2
}
]
}
]
}
|
| Gold 51 | \mathcal{K}_k(x; n,q) = \sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \binom {n-j}{k-j} \binom{x}{j}
|
Kravchuk polynomials |
|
|
- | - | - |
|
- | Krawtchouk vs Kravchuk (synonym problem) | Full data:
{
"id": 51,
"pid": 102,
"eid": "math.102.5",
"title": "Kravchuk polynomials",
"formulae": [
{
"id": "FORMULA_6b7eb62a3e02e45fb1365dd2f07a5bbc",
"formula": "\\mathcal{K}_k(x; n,q) = \\sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \\binom {n-j}{k-j} \\binom{x}{j}",
"semanticFormula": "\\KrawtchoukpolyK{k}@{x}{n}{q} = \\sum_{j=0}^{k}(-q)^j (q-1)^{k-j} \\binom {n-j}{k-j} \\binom{x}{j}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "K[k_, x_, n_, q_] := Sum[(- q)^(j)*(q - 1)^(k - j)*Binomial[n - j,k - j]*Binomial[x,j], {j, 0, k}, GenerateConditions->None]"
}
},
"positions": [
{
"section": 2,
"sentence": 0,
"word": 9
}
],
"includes": [
"q",
"n"
],
"isPartOf": [],
"definiens": [
{
"definition": "following alternative expression",
"score": 0
},
{
"definition": "Kravchuk polynomial",
"score": 2
}
]
}
]
} |
| Gold 52 | g_1(x) = \sum_{k \geq 1} \frac{\sin(k \pi / 4)}{k! (8x)^k} \prod_{l = 1}^k (2l - 1)^2
|
Kelvin functions |
|
|
|
- | - | - | - | - | Full data:
{
"id": 52,
"pid": 103,
"eid": "math.103.8",
"title": "Kelvin functions",
"formulae": [
{
"id": "FORMULA_07453e6baf8f216467f9b664de795bfc",
"formula": "g_1(x) = \\sum_{k \\geq 1} \\frac{\\sin(k \\pi \/ 4)}{k! (8x)^k} \\prod_{l = 1}^k (2l - 1)^2",
"semanticFormula": "g_1(x) = \\sum_{k \\geq 1} \\frac{\\sin(k \\cpi \/ 4)}{k! (8x)^k} \\prod_{l = 1}^k(2 l - 1)^2",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Subscript[g, 1][x_] := Sum[Divide[Sin[k*Pi\/4],(k)!*(8*x)^(k)]*Product[(2*l - 1)^(2), {l, 1, k}, GenerateConditions->None], {k, 1, Infinity}, GenerateConditions->None]"
},
"Maple": {
"translation": "g[1] := (x) -> sum((sin(k*Pi\/4))\/(factorial(k)*(8*x)^(k))*product((2*l - 1)^(2), l = 1..k), k = 1..infinity)"
}
},
"positions": [
{
"section": 1,
"sentence": 1,
"word": 28
}
],
"includes": [
"x",
"x)",
"g_1(x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "series expansion",
"score": 1
},
{
"definition": "special case",
"score": 0
},
{
"definition": "asymptotic series",
"score": 1
},
{
"definition": "definition",
"score": 2
}
]
}
]
} |
| Gold 53 | S_{\mu,\nu}(z) = s_{\mu,\nu}(z) + 2^{\mu-1} \Gamma\left(\frac{\mu + \nu + 1}{2}\right) \Gamma\left(\frac{\mu - \nu + 1}{2}\right)\left(\sin \left[(\mu - \nu)\frac{\pi}{2}\right] J_\nu(z) - \cos \left[(\mu - \nu)\frac{\pi}{2}\right] Y_\nu(z)\right)
|
Lommel function |
|
|
- | - | - |
|
- | No information about gamma function | Full data:
{
"id": 53,
"pid": 104,
"eid": "math.104.2",
"title": "Lommel function",
"formulae": [
{
"id": "FORMULA_03f5cb50caaedb9f0a4ada231fd61c58",
"formula": "S_{\\mu,\\nu}(z) = s_{\\mu,\\nu}(z) + 2^{\\mu-1} \\Gamma\\left(\\frac{\\mu + \\nu + 1}{2}\\right) \\Gamma\\left(\\frac{\\mu - \\nu + 1}{2}\\right)\\left(\\sin \\left[(\\mu - \\nu)\\frac{\\pi}{2}\\right] J_\\nu(z) - \\cos \\left[(\\mu - \\nu)\\frac{\\pi}{2}\\right] Y_\\nu(z)\\right)",
"semanticFormula": "\\LommelS{\\mu}{\\nu}@{z} = \\Lommels{\\mu}{\\nu}@{z} + 2^{\\mu-1} \\EulerGamma@{\\frac{\\mu + \\nu + 1}{2}} \\EulerGamma@{\\frac{\\mu - \\nu + 1}{2}}(\\sin [(\\mu - \\nu) \\frac{\\cpi}{2}] \\BesselJ{\\nu}@{z} - \\cos [(\\mu - \\nu) \\frac{\\cpi}{2}] \\BesselY{\\nu}@{z})",
"confidence": 0.8775479393290169,
"translations": {
"Mathematica": {
"translation": "S[\\[Mu]_, \\[Nu]_, z_] := Divide[Pi,2]*(BesselY[\\[Nu], z]*Integrate[(x)^\\[Mu]* BesselJ[\\[Nu], x], {x, 0, z}, GenerateConditions->None]- BesselJ[\\[Nu], z]*Integrate[(x)^\\[Mu]* BesselY[\\[Nu], x], {x, 0, z}, GenerateConditions->None]) + (2)^(\\[Mu]- 1)* Gamma[Divide[\\[Mu]+ \\[Nu]+ 1,2]]*Gamma[Divide[\\[Mu]- \\[Nu]+ 1,2]]*(Sin[((\\[Mu]- \\[Nu])*Divide[Pi,2])*]*BesselJ[\\[Nu], z]- Cos[((\\[Mu]- \\[Nu])*Divide[Pi,2])*]*BesselY[\\[Nu], z])"
},
"Maple": {
"translation": "LommelS1(mu, nu, z) = (Pi)\/(2)*(BesselY(nu, z)*int((x)^(mu)* BesselJ(nu, x), x = 0..z)- BesselJ(nu, z)*int((x)^(mu)* BesselY(nu, x), x = 0..z))"
}
},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 18
}
],
"includes": [
"s_{\\mu,\\nu}(z)",
"S_{\\mu,\\nu}(z)",
"J_{\\nu}(z)",
"Y_{\\nu}(z)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Lommel function",
"score": 2
},
{
"definition": "Bessel function of the first kind",
"score": 2
},
{
"definition": "Bessel function of the second kind",
"score": 2
}
]
}
]
} |
| Gold 54 | \mathbf{H}_{\alpha}(z) = \frac{z^{\alpha+1}}{2^{\alpha}\sqrt{\pi} \Gamma \left (\alpha+\tfrac{3}{2} \right )} {}_1F_2 \left (1,\tfrac{3}{2}, \alpha+\tfrac{3}{2},-\tfrac{z^2}{4} \right )
|
Struve function |
|
|
- |
|
- | - | - | Arguments of are split by commas. That is wrong notation. Hence, our semantic patterns did not match. | Full data:
{
"id": 54,
"pid": 105,
"eid": "math.105.18",
"title": "Struve function",
"formulae": [
{
"id": "FORMULA_6dc2da7f595d2f199fbc15768167f006",
"formula": "\\mathbf{H}_{\\alpha}(z) = \\frac{z^{\\alpha+1}}{2^{\\alpha}\\sqrt{\\pi} \\Gamma \\left (\\alpha+\\tfrac{3}{2} \\right )} {}_1F_2 \\left (1,\\tfrac{3}{2}, \\alpha+\\tfrac{3}{2},-\\tfrac{z^2}{4} \\right )",
"semanticFormula": "\\StruveH{\\alpha}@{z} = \\frac{z^{\\alpha+1}}{2^{\\alpha} \\sqrt{\\cpi} \\EulerGamma@{\\alpha + \\tfrac{3}{2}}} \\genhyperF{1}{2}@{1}{\\tfrac{3}{2}, \\alpha + \\tfrac{3}{2}}{- \\tfrac{z^2}{4}}",
"confidence": 0.8740850655136605,
"translations": {
"Mathematica": {
"translation": "StruveH[\\[Alpha], z] == Divide[(z)^(\\[Alpha]+ 1),(2)^\\[Alpha]*Sqrt[Pi]*Gamma[\\[Alpha]+Divide[3,2]]]*HypergeometricPFQ[{1}, {Divide[3,2], \\[Alpha]+Divide[3,2]}, -Divide[(z)^(2),4]]"
},
"Maple": {
"translation": "StruveH(alpha, z) = ((z)^(alpha + 1))\/((2)^(alpha)*sqrt(Pi)*GAMMA(alpha +(3)\/(2)))*hypergeom([1], [(3)\/(2), alpha +(3)\/(2)], -((z)^(2))\/(4))"
}
},
"positions": [
{
"section": 6,
"sentence": 2,
"word": 31
}
],
"includes": [
"_{1}F_{2}",
"\\mathbf{K}_\\alpha(x)",
"\\alpha",
"\\Gamma(z)",
"\\mathbf{H}_{\\alpha}(x)",
"\\mathbf{L}_{\\alpha}(x)",
"\\mathbf{H}_{\\alpha}(z)",
"Y_{\\alpha}(x)",
"\\mathbf{M}_\\alpha(x)"
],
"isPartOf": [],
"definiens": [
{
"definition": "hypergeometric function",
"score": 2
},
{
"definition": "Struve",
"score": 2
},
{
"definition": "Struve function",
"score": 2
},
{
"definition": "gamma function",
"score": 2
}
]
}
]
} |
| Gold 55 | f(t+p) = f(t)
|
Hill differential equation |
|
|
- | - | - | - | - | - | Full data:
{
"id": 55,
"pid": 106,
"eid": "math.106.7",
"title": "Hill differential equation",
"formulae": [
{
"id": "FORMULA_3a6745862e8f6ef2b93c343ad82b40c0",
"formula": "f(t+p) = f(t)",
"semanticFormula": "f(t+p) = f(t)",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "f[t + p] == f[t]"
},
"Maple": {
"translation": "f(t + p) = f(t)"
}
},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 21
}
],
"includes": [
"f(t)",
"t",
"p",
"f(t+\\pi)=f(t)"
],
"isPartOf": [
"f(t+\\pi)=f(t)"
],
"definiens": [
{
"definition": "function",
"score": 2
},
{
"definition": "periodic function by minimal period",
"score": 2
}
]
}
]
} |
| Gold 56 | \mathbf{J}_\nu(z)=\cos\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k}}{4^k\Gamma\left(k+\frac{\nu}{2}+1\right)\Gamma\left(k-\frac{\nu}{2}+1\right)}+\sin\frac{\pi\nu}{2}\sum_{k=0}^\infty\frac{(-1)^kz^{2k+1}}{2^{2k+1}\Gamma\left(k+\frac{\nu}{2}+\frac{3}{2}\right)\Gamma\left(k-\frac{\nu}{2}+\frac{3}{2}\right)}
|
Anger function |
|
|
- | - | - |
|
- | No information about gamma function. | Full data:
{
"id": 56,
"pid": 108,
"eid": "math.108.3",
"title": "Anger function",
"formulae": [
{
"id": "FORMULA_014efde25f995ccd08168a36ec7ef86d",
"formula": "\\mathbf{J}_\\nu(z)=\\cos\\frac{\\pi\\nu}{2}\\sum_{k=0}^\\infty\\frac{(-1)^kz^{2k}}{4^k\\Gamma\\left(k+\\frac{\\nu}{2}+1\\right)\\Gamma\\left(k-\\frac{\\nu}{2}+1\\right)}+\\sin\\frac{\\pi\\nu}{2}\\sum_{k=0}^\\infty\\frac{(-1)^kz^{2k+1}}{2^{2k+1}\\Gamma\\left(k+\\frac{\\nu}{2}+\\frac{3}{2}\\right)\\Gamma\\left(k-\\frac{\\nu}{2}+\\frac{3}{2}\\right)}",
"semanticFormula": "\\AngerJ{\\nu}@{z} = \\cos \\frac{\\cpi\\nu}{2} \\sum_{k=0}^\\infty \\frac{(-1)^k z^{2k}}{4^k\\EulerGamma@{k+\\frac{\\nu}{2}+1}\\EulerGamma@{k-\\frac{\\nu}{2}+1}}+\\sin\\frac{\\cpi\\nu}{2}\\sum_{k=0}^\\infty\\frac{(-1)^k z^{2k+1}}{2^{2k+1}\\EulerGamma@{k+\\frac{\\nu}{2}+\\frac{3}{2}}\\EulerGamma@{k-\\frac{\\nu}{2}+\\frac{3}{2}}}",
"confidence": 0.8648813564530858,
"translations": {
"Mathematica": {
"translation": "AngerJ[\\[Nu], z] == Cos[Divide[Pi*\\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k),(4)^(k)* Gamma[k +Divide[\\[Nu],2]+ 1]*Gamma[k -Divide[\\[Nu],2]+ 1]], {k, 0, Infinity}, GenerateConditions->None]+ Sin[Divide[Pi*\\[Nu],2]]*Sum[Divide[(- 1)^(k)* (z)^(2*k + 1),(2)^(2*k + 1)* Gamma[k +Divide[\\[Nu],2]+Divide[3,2]]*Gamma[k -Divide[\\[Nu],2]+Divide[3,2]]], {k, 0, Infinity}, GenerateConditions->None]"
},
"Maple": {
"translation": "AngerJ(nu, z) = cos((Pi*nu)\/(2))*sum(((- 1)^(k)* (z)^(2*k))\/((4)^(k)* GAMMA(k +(nu)\/(2)+ 1)*GAMMA(k -(nu)\/(2)+ 1)), k = 0..infinity)+ sin((Pi*nu)\/(2))*sum(((- 1)^(k)* (z)^(2*k + 1))\/((2)^(2*k + 1)* GAMMA(k +(nu)\/(2)+(3)\/(2))*GAMMA(k -(nu)\/(2)+(3)\/(2))), k = 0..infinity)"
}
},
"positions": [
{
"section": 2,
"sentence": 0,
"word": 8
}
],
"includes": [
"J_{\\nu}",
"\\mathbf{J}_{\\nu}",
"\\nu"
],
"isPartOf": [],
"definiens": [
{
"definition": "power series expansion",
"score": 2
},
{
"definition": "Anger function",
"score": 2
},
{
"definition": "Gamma function",
"score": 2
}
]
}
]
} |
| Gold 57 | (\operatorname{Ec})^'_{2K} = (\operatorname{Ec})^'_0 = 0, \;\; (\operatorname{Es})^'_{2K} = (\operatorname{Es})^'_0 = 0
|
Lamé function |
|
- | - | - | - | - | - | No translation possible. | Full data:
{
"id": 57,
"pid": 109,
"eid": "math.109.27",
"title": "Lam\u00e9 function",
"formulae": [
{
"id": "FORMULA_7d20395e75eeb74df48a681897d9d727",
"formula": "(\\operatorname{Ec})^'_{2K} = (\\operatorname{Ec})^'_0 = 0, \\;\\; (\\operatorname{Es})^'_{2K} = (\\operatorname{Es})^'_0 = 0",
"semanticFormula": "(\\operatorname{Ec})_{2K}^' =(\\operatorname{Ec})_0^' = 0 ,(\\operatorname{Es})_{2K}^' =(\\operatorname{Es})_0^' = 0",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 2,
"sentence": 3,
"word": 31
}
],
"includes": [
"\\operatorname{Ec}",
"\\operatorname{Es}"
],
"isPartOf": [],
"definiens": [
{
"definition": "boundary condition",
"score": 2
},
{
"definition": "ellipsoidal wave",
"score": 2
}
]
}
]
} |
| Gold 58 | \int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx \approx \sum_{i=1}^n w_i f(x_i)
|
Gauss–Hermite quadrature |
|
- | - | - | - | - | - | No translation possible. | Full data:
{
"id": 58,
"pid": 110,
"eid": "math.110.1",
"title": "Gauss\u2013Hermite quadrature",
"formulae": [
{
"id": "FORMULA_cdf8d887d4b5ad1a7724773d8eef8fd2",
"formula": "\\int_{-\\infty}^{+\\infty} e^{-x^2} f(x)\\,dx \\approx \\sum_{i=1}^n w_i f(x_i)",
"semanticFormula": "\\int_{-\\infty}^{+\\infty} \\expe^{-x^2} f(x) \\diff{x} \\approx \\sum_{i=1}^n w_i f(x_i)",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 3
}
],
"includes": [
"\\int_{-\\infty}^{+\\infty} e^{-x^2} f(x)\\,dx",
"n",
"x_{i}",
"w_{i}"
],
"isPartOf": [],
"definiens": [
{
"definition": "value of integral",
"score": 2
},
{
"definition": "form of Gaussian quadrature",
"score": 2
},
{
"definition": "Gauss -- Hermite quadrature",
"score": 2
},
{
"definition": "Hermite polynomial",
"score": 1
},
{
"definition": "associated weight",
"score": 2
}
]
}
]
} |
| Gold 59 | p_n(x;a,b,c,d|q) =(ab,ac,ad;q)_na^{-n}\;_{4}\phi_3 \left[\begin{matrix} q^{-n}&abcdq^{n-1}&ae^{i\theta}&ae^{-i\theta} \\ ab&ac&ad \end{matrix} ; q,q \right]
|
Askey–Wilson polynomials |
|
|
|
- | - | - | - | Could not extract the name Askey-Wilson polynomials. | Full data:
{
"id": 59,
"pid": 111,
"eid": "math.111.0",
"title": "Askey\u2013Wilson polynomials",
"formulae": [
{
"id": "FORMULA_cfe946a0547913234ac79d398f269607",
"formula": "p_n(x;a,b,c,d|q) =(ab,ac,ad;q)_na^{-n}\\;_{4}\\phi_3 \\left[\\begin{matrix} q^{-n}&abcdq^{n-1}&ae^{i\\theta}&ae^{-i\\theta} \\\\ ab&ac&ad \\end{matrix} ; q,q \\right]",
"semanticFormula": "\\AskeyWilsonpolyp{n}@{x}{a}{b}{c}{d}{q} = \\qmultiPochhammersym{ab , ac , ad}{q}{n} a^{-n} \\qgenhyperphi{4}{3}@{q^{-n} , abcdq^{n-1} , a\\expe^{\\iunit\\theta} , a\\expe^{-\\iunit\\theta}}{ab , ac , ad}{q}{q}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "p[n_, x_, a_, b_, c_, d_, q_] := Product[QPochhammer[Part[{a*b , a*c , a*d},i],q,n],{i,1,Length[{a*b , a*c , a*d}]}]*(a)^(- n)* QHypergeometricPFQ[{(q)^(- n), a*b*c*d*(q)^(n - 1), a*Exp[I*\\[Theta]], a*Exp[- I*\\[Theta]]},{a*b , a*c , a*d},q,q]"
}
},
"positions": [
{
"section": 0,
"sentence": 3,
"word": 4
}
],
"includes": [
"\\phi",
"_{n}",
"n"
],
"isPartOf": [],
"definiens": [
{
"definition": "basic hypergeometric function",
"score": 2
},
{
"definition": "q-Pochhammer symbol",
"score": 2
},
{
"definition": "Askey\u2013Wilson polynomials",
"score": 2
}
]
}
]
} |
| Gold 60 | Q_n(x;\alpha,\beta,N)= {}_3F_2(-n,-x,n+\alpha+\beta+1;\alpha+1,-N+1;1).
|
Hahn polynomials |
|
|
|
- | - | - |
|
- | Full data:
{
"id": 60,
"pid": 112,
"eid": "math.112.0",
"title": "Hahn polynomials",
"formulae": [
{
"id": "FORMULA_777007203448847310455e0b0eaaeb2c",
"formula": "Q_n(x;\\alpha,\\beta,N)= {}_3F_2(-n,-x,n+\\alpha+\\beta+1;\\alpha+1,-N+1;1).",
"semanticFormula": "\\HahnpolyQ{n}@{x}{\\alpha}{\\beta}{N} = \\genhyperF{3}{2}@{- n , - x , n + \\alpha + \\beta + 1}{\\alpha + 1 , - N + 1}{1}",
"confidence": 0.8953028732079359,
"translations": {
"Mathematica": {
"translation": "Q[n_, x_, \\[Alpha]_, \\[Beta]_, N_] := HypergeometricPFQ[{- n , - x , n + \\[Alpha]+ \\[Beta]+ 1}, {\\[Alpha]+ 1 , - N + 1}, 1]"
}
},
"positions": [
{
"section": 0,
"sentence": 3,
"word": 11
}
],
"includes": [
"R_{n}(x;\\gamma,\\delta,N)",
"S_{n}(x;a,b,c)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Hahn polynomial",
"score": 2
},
{
"definition": "basic hypergeometric function",
"score": 2
},
{
"definition": "hypergeometric function",
"score": 2
}
]
}
]
} |
| Gold 61 | \sum_{x=0}^\infty \frac{\mu^x}{x!} C_n(x; \mu)C_m(x; \mu)=\mu^{-n} e^\mu n! \delta_{nm}, \quad \mu>0
|
Charlier polynomials |
|
|
- | - | - |
|
- | Did not found Charlier polynomial. | Full data:
{
"id": 61,
"pid": 113,
"eid": "math.113.2",
"title": "Charlier polynomials",
"formulae": [
{
"id": "FORMULA_b76bcf7237b989f6b5d90082fafa53f1",
"formula": "\\sum_{x=0}^\\infty \\frac{\\mu^x}{x!} C_n(x; \\mu)C_m(x; \\mu)=\\mu^{-n} e^\\mu n! \\delta_{nm}, \\quad \\mu>0",
"semanticFormula": "\\sum_{x=0}^\\infty \\frac{\\mu^x}{x!} \\CharlierpolyC{n}@{x}{\\mu} \\CharlierpolyC{m}@{x}{\\mu} = \\mu^{-n} \\expe^\\mu n! \\delta_{nm} , \\quad \\mu > 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Sum[Divide[\\[Mu]^x, x!] * HypergeometricPFQ[{-n, -x}, {}, -Divide[1,\\[Mu]]] * HypergeometricPFQ[{-m, -x}, {}, -Divide[1,\\[Mu]]], {x, 0, Infinity}] == \\[Mu]^(-n)*Exp[\\[Mu]]*n!*Subscript[\\[Delta], n, m]"
}
},
"positions": [
{
"section": 0,
"sentence": 2,
"word": 5
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "orthogonality relation",
"score": 2
},
{
"definition": "Charlier polynomial",
"score": 2
}
]
}
]
} |
| Gold 62 | p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\phi_3\left[\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\\aq&bdq&cq\\ \end{matrix};q;q\right]
|
Q-Racah polynomials |
|
- | - | - | - | - |
|
Did not find q-Recah polynomial. Since it is not a definition, and q-Recah are not supported by Mathematica, there is no translation possible. | Full data:
{
"id": 62,
"pid": 114,
"eid": "math.114.0",
"title": "Q-Racah polynomials",
"formulae": [
{
"id": "FORMULA_51c23bddc19530680328afbf28235b90",
"formula": "p_n(q^{-x}+q^{x+1}cd;a,b,c,d;q) = {}_4\\phi_3\\left[\\begin{matrix} q^{-n} &abq^{n+1}&q^{-x}&q^{x+1}cd\\\\aq&bdq&cq\\\\ \\end{matrix};q;q\\right]",
"semanticFormula": "\\qRacahpolyR{n}@{q^{-x} + q^{x+1} cd}{a}{b}{c}{d}{q} = \\qgenhyperphi{4}{3}@{q^{-n}, abq^{n+1}, q^{-x}, q^{x+1}cd}{aq , bdq , cq}{q}{q}",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "term of basic hypergeometric function",
"score": 2
}
]
}
]
} |
| Gold 63 | \displaystyle c_n(q^{-x};a;q) = {}_2\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}/a)
|
Q-Charlier polynomials |
|
- | - | - | - | - | - |
|
Full data:
{
"id": 63,
"pid": 115,
"eid": "math.115.0",
"title": "Q-Charlier polynomials",
"formulae": [
{
"id": "FORMULA_925d68ff3ddf733a69ec9936dfede5d6",
"formula": "\\displaystyle c_n(q^{-x};a;q) = {}_2\\phi_1(q^{-n},q^{-x};0;q,-q^{n+1}\/a)",
"semanticFormula": "c_n(q^{-x} ; a ; q) = \\qgenhyperphi{2}{1}@{q^{-n} , q^{-x}}{0}{q}{- q^{n+1} \/ a}",
"confidence": 0.5776294951318733,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "q-Charlier polynomial",
"score": 2
},
{
"definition": "term of the basic hypergeometric function",
"score": 2
}
]
}
]
} |
| Gold 64 | M_n(x,\beta,\gamma) = \sum_{k=0}^n (-1)^k{n \choose k}{x\choose k}k!(x+\beta)_{n-k}\gamma^{-k}
|
Meixner polynomials |
|
|
- | - | - |
|
- | Did not find Meixner. | Full data:
{
"id": 64,
"pid": 116,
"eid": "math.116.0",
"title": "Meixner polynomials",
"formulae": [
{
"id": "FORMULA_29a1f82de004c5721c8dfc5dd1dc5b98",
"formula": "M_n(x,\\beta,\\gamma) = \\sum_{k=0}^n (-1)^k{n \\choose k}{x\\choose k}k!(x+\\beta)_{n-k}\\gamma^{-k}",
"semanticFormula": "\\MeixnerpolyM{n}@{x}{\\beta}{\\gamma} = \\sum_{k=0}^n(- 1)^k{n \\choose k}{x\\choose k} k! \\Pochhammersym{x + \\beta}{n-k} \\gamma^{-k}",
"confidence": 0.8953028732079359,
"translations": {
"Mathematica": {
"translation": "M[n_, x_, \\[Beta]_, \\[Gamma]_] := Sum[(- 1)^(k)*Binomial[n,k]*Binomial[x,k]*(k)!*Pochhammer[x + \\[Beta], n - k]*\\[Gamma]^(- k), {k, 0, n}, GenerateConditions->None]"
}
},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 16
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "Meixner polynomial",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "term of binomial coefficient",
"score": 1
}
]
}
]
} |
| Gold 65 | x(1-x) \frac {\partial^2F_1(x,y)} {\partial x^2} + y(1-x) \frac {\partial^2F_1(x,y)} {\partial x \partial y} + [c - (a+b_1+1) x] \frac {\partial F_1(x,y)} {\partial x} - b_1 y \frac {\partial F_1(x,y)} {\partial y} - a b_1 F_1(x,y) = 0
|
Appell series |
|
|
- |
|
- |
|
- | Cannot match hidden arguments of Appell F1 function. | Full data:
{
"id": 65,
"pid": 117,
"eid": "math.117.19",
"title": "Appell series",
"formulae": [
{
"id": "FORMULA_85014aaf0c7c1f4fe433115e796a03db",
"formula": "x(1-x) \\frac {\\partial^2F_1(x,y)} {\\partial x^2} + y(1-x) \\frac {\\partial^2F_1(x,y)} {\\partial x \\partial y} + [c - (a+b_1+1) x] \\frac {\\partial F_1(x,y)} {\\partial x} - b_1 y \\frac {\\partial F_1(x,y)} {\\partial y} - a b_1 F_1(x,y) = 0",
"semanticFormula": "x(1-x) \\deriv[2]{\\AppellF{1}@{a}{b_1}{b_2}{\\gamma}{x}{y}}{x} + y(1-x) \\frac{\\pdiff[2]{\\AppellF{1}@{a}{b_1}{b_2}{\\gamma}{x}{y}}}{\\pdiff{x}\\pdiff{y}} + [c - (a+b_1+1) x] \\deriv[1]{\\AppellF{1}@{a}{b_1}{b_2}{\\gamma}{x}{y}}{x} - b_1 y \\deriv[1]{\\AppellF{1}@{a}{b_1}{b_2}{\\gamma}{x}{y}}{y} - a b_1 \\AppellF{1}@{a}{b_1}{b_2}{\\gamma}{x}{y} = 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "x*(1-x) * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \\[Gamma], x, y], {x,2}] + y*(1-x) * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \\[Gamma], x, y], x, y] + (c - (a+Subscript[b, 1]+1)*x) * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \\[Gamma], x, y], x] - Subscript[b,1] * y * D[AppellF[a, Subscript[b, 1], Subscript[b, 2], \\[Gamma], x, y], y] - a*Subscript[b,1]*AppellF[a, Subscript[b, 1], Subscript[b, 2], \\[Gamma], x, y] == 0"
}
},
"positions": [
{
"section": 3,
"sentence": 0,
"word": 39
}
],
"includes": [
"y",
"x",
"F_{1}",
"F",
"_{1}F_{1}"
],
"isPartOf": [],
"definiens": [
{
"definition": "Appell",
"score": 2
},
{
"definition": "partial differential equation",
"score": 2
},
{
"definition": "system of differential equation",
"score": 1
},
{
"definition": "system of second-order differential equation",
"score": 2
},
{
"definition": "Appell series",
"score": 2
}
]
}
]
} |
| Gold 66 | \Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau > 0
|
Theta function of a lattice |
|
|
|
- | - | - | - | - | Full data:
{
"id": 66,
"pid": 118,
"eid": "math.118.0",
"title": "Theta function of a lattice",
"formulae": [
{
"id": "FORMULA_39f4baaa3543f22706b6f7701518f3eb",
"formula": "\\Theta_\\Lambda(\\tau) = \\sum_{x\\in\\Lambda}e^{i\\pi\\tau\\|x\\|^2}\\qquad\\mathrm{Im}\\,\\tau > 0",
"semanticFormula": "\\Theta_\\Lambda(\\tau) = \\sum_{x\\in\\Lambda} \\expe^{\\iunit \\cpi \\tau \\|x \\|^2} \\qquad \\imagpart \\tau > 0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "\\[CapitalTheta][\\[CapitalLambda]_, \\[Tau]_] := Sum[Exp[I*Pi*\\[Tau]*(Norm[x])^(2)], {x, \\[CapitalLambda]}]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"\\Lambda"
],
"isPartOf": [],
"definiens": [
{
"definition": "theta function",
"score": 2
},
{
"definition": "lattice",
"score": 1
},
{
"definition": "Theta function of a lattice",
"score": 1
}
]
}
]
} |
| Gold 67 | \frac{d^2 S}{dz^2}+\left(\sum _{j=1}^N \frac{\gamma _j}{z - a_j} \right) \frac{dS}{dz} + \frac{V(z)}{\prod _{j=1}^N (z - a_j)}S = 0
|
Heine–Stieltjes polynomials |
|
- | - |
|
- | - | - | Mistakenly detected Stieltjes constant. No translation possible for S. | Full data:
{
"id": 67,
"pid": 119,
"eid": "math.119.0",
"title": "Heine\u2013Stieltjes polynomials",
"formulae": [
{
"id": "FORMULA_d673cd2334542e8f83f099798c4027b3",
"formula": "\\frac{d^2 S}{dz^2}+\\left(\\sum _{j=1}^N \\frac{\\gamma _j}{z - a_j} \\right) \\frac{dS}{dz} + \\frac{V(z)}{\\prod _{j=1}^N (z - a_j)}S = 0",
"semanticFormula": "\\deriv [2]{S}{z} +(\\sum_{j=1}^N \\frac{\\gamma _j}{z - a_j}) \\deriv[]{S}{z} + \\frac{V(z)}{\\prod _{j=1}^N (z - a_j)} S = 0",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 6
}
],
"includes": [
"V(z)",
"S",
"V"
],
"isPartOf": [],
"definiens": [
{
"definition": "form",
"score": 0
},
{
"definition": "Fuchsian equation",
"score": 2
},
{
"definition": "polynomial",
"score": 1
},
{
"definition": "degree",
"score": 0
},
{
"definition": "Edward Burr Van Vleck",
"score": 0
},
{
"definition": "Heine",
"score": 1
},
{
"definition": "polynomial solution",
"score": 1
},
{
"definition": "Stieltjes polynomial",
"score": 1
},
{
"definition": "Van Vleck polynomial",
"score": 1
}
]
}
]
} |
| Gold 68 | w(x) = \frac{k}{\sqrt{\pi}} x^{-1/2} \exp(-k^2\log^2 x)
|
Stieltjes–Wigert polynomials |
|
|
|
- | - | - | - | - | Full data:
{
"id": 68,
"pid": 120,
"eid": "math.120.0",
"title": "Stieltjes\u2013Wigert polynomials",
"formulae": [
{
"id": "FORMULA_583d3b9e00bbd73091b01f368d1a82c7",
"formula": "w(x) = \\frac{k}{\\sqrt{\\pi}} x^{-1\/2} \\exp(-k^2\\log^2 x)",
"semanticFormula": "w(x) = \\frac{k}{\\sqrt{\\cpi}} x^{-1\/2} \\exp(- k^2 \\log^2 x)",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "w[x_] := Divide[k,Sqrt[Pi]]*(x)^(- 1\/2)* Exp[- (k)^(2)* (Log[x])^(2)]"
},
"Maple": {
"translation": "w := (x) -> (k)\/(sqrt(Pi))*(x)^(- 1\/2)* exp(- (k)^(2)* (log(x))^(2))"
}
},
"positions": [
{
"section": 0,
"sentence": 0,
"word": 38
}
],
"includes": [
"\\frac{k}{\\sqrt{\\pi}} x^{-1\/2} \\exp \\left(-k^2 \\log^2 x \\right)"
],
"isPartOf": [],
"definiens": [
{
"definition": "weight function",
"score": 2
},
{
"definition": "positive real line",
"score": 0
},
{
"definition": "basic Askey scheme",
"score": 1
},
{
"definition": "family of basic hypergeometric orthogonal polynomial",
"score": 1
},
{
"definition": "mathematics",
"score": 0
},
{
"definition": "Stieltjes -- Wigert polynomial",
"score": 2
},
{
"definition": "Thomas Jan Stieltjes",
"score": 0
},
{
"definition": "Carl Severin Wigert",
"score": 0
},
{
"definition": "example of such weight function",
"score": 0
}
]
}
]
} |
| Gold 69 | y^2=x(x-1)(x-\lambda)
|
Modular lambda function |
|
|
- | - | - | - | - | - | Full data:
{
"id": 69,
"pid": 121,
"eid": "math.121.23",
"title": "Modular lambda function",
"formulae": [
{
"id": "FORMULA_4e5334aa6f5fa551b0718a2372816061",
"formula": "y^2=x(x-1)(x-\\lambda)",
"semanticFormula": "y^2=x(x-1)(x-\\lambda)",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "(y)^(2) == x*(x - 1)*(x - \\[Lambda])"
},
"Maple": {
"translation": "(y)^(2) = x*(x - 1)*(x - lambda)"
}
},
"positions": [
{
"section": 2,
"sentence": 5,
"word": 13
}
],
"includes": [
"\\lambda"
],
"isPartOf": [],
"definiens": [
{
"definition": "elliptic curve of Legendre form",
"score": 2
},
{
"definition": "relation to the j-invariant",
"score": 1
},
{
"definition": "relation to the j-invariant",
"score": 1
}
]
}
]
} |
| Gold 70 | P_1^{(\lambda)}(x;\phi)=2(\lambda\cos\phi + x\sin\phi)
|
Meixner–Pollaczek polynomials |
|
- | - | - | - | - |
|
- | Full data:
{
"id": 70,
"pid": 122,
"eid": "math.122.3",
"title": "Meixner\u2013Pollaczek polynomials",
"formulae": [
{
"id": "FORMULA_96d19b4b504f801548c69064d662043b",
"formula": "P_1^{(\\lambda)}(x;\\phi)=2(\\lambda\\cos\\phi + x\\sin\\phi)",
"semanticFormula": "\\MeixnerPollaczekpolyP{\\lambda}{1}@{x}{\\phi} = 2(\\lambda \\cos \\phi + x \\sin \\phi)",
"confidence": 0.8953028732079359,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 9
}
],
"includes": [
"P_{m}^{(\\lambda)}(x;\\varphi)"
],
"isPartOf": [],
"definiens": [
{
"definition": "first few Meixner -- Pollaczek polynomial",
"score": 2
}
]
}
]
} |
| Gold 71 | P_n^{(\alpha,\beta)}(z)=\frac{(\alpha+1)_n}{n!}\,{}_2F_1\left(-n,1+\alpha+\beta+n;\alpha+1;\tfrac{1}{2}(1-z)\right)
|
Jacobi polynomials |
|
|
- | - | - | - | - | - | Full data:
{
"id": 71,
"pid": 123,
"eid": "math.123.0",
"title": "Jacobi polynomials",
"formulae": [
{
"id": "FORMULA_c8b5b9184e45bca39744427c45693115",
"formula": "P_n^{(\\alpha,\\beta)}(z)=\\frac{(\\alpha+1)_n}{n!}\\,{}_2F_1\\left(-n,1+\\alpha+\\beta+n;\\alpha+1;\\tfrac{1}{2}(1-z)\\right)",
"semanticFormula": "\\JacobipolyP{\\alpha}{\\beta}{n}@{z} = \\frac{\\Pochhammersym{\\alpha + 1}{n}}{n!} \\genhyperF{2}{1}@{- n , 1 + \\alpha + \\beta + n}{\\alpha + 1}{\\tfrac{1}{2}(1 - z)}",
"confidence": 0.7595006538205181,
"translations": {
"Mathematica": {
"translation": "JacobiP[n, \\[Alpha], \\[Beta], z] == Divide[Pochhammer[\\[Alpha]+ 1, n],(n)!]*HypergeometricPFQ[{- n , 1 + \\[Alpha]+ \\[Beta]+ n}, {\\[Alpha]+ 1}, Divide[1,2]*(1 - z)]"
},
"Maple": {
"translation": "JacobiP(n, alpha, beta, z) = (pochhammer(alpha + 1, n))\/(factorial(n))*hypergeom([- n , 1 + alpha + beta + n], [alpha + 1], (1)\/(2)*(1 - z))"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 12
}
],
"includes": [
"P_{n}^{(\\alpha, \\beta)}(x)",
"(\\alpha+1)_n",
"n",
"n + \\alpha + \\beta",
"P_{n}^{(\\alpha, \\beta)}",
"\\alpha,\\beta",
"z"
],
"isPartOf": [],
"definiens": [
{
"definition": "Pochhammer 's symbol",
"score": 2
},
{
"definition": "hypergeometric function",
"score": 2
},
{
"definition": "Jacobi polynomial",
"score": 2
}
]
}
]
} |
| Gold 72 | S_n(x^2;a,b,c)= {}_3F_2(-n,a+ix,a-ix;a+b,a+c;1).
|
Continuous dual Hahn polynomials |
|
- | - | - | - | - |
|
- | Full data:
{
"id": 72,
"pid": 124,
"eid": "math.124.0",
"title": "Continuous dual Hahn polynomials",
"formulae": [
{
"id": "FORMULA_b0d448ba925dc6b2bf2ce32a1253dee4",
"formula": "S_n(x^2;a,b,c)= {}_3F_2(-n,a+ix,a-ix;a+b,a+c;1).",
"semanticFormula": "\\contdualHahnpolyS{n}@{x^2}{a}{b}{c} = \\genhyperF{3}{2}@{- n , a + \\iunit x , a - \\iunit x}{a + b , a + c}{1}",
"confidence": 0.7132263353695951,
"translations": {},
"positions": [
{
"section": 0,
"sentence": 1,
"word": 10
}
],
"includes": [
"R_{n}(x;\\gamma,\\delta,N)"
],
"isPartOf": [],
"definiens": [
{
"definition": "hypergeometric function",
"score": 1
},
{
"definition": "dual Hahn polynomial",
"score": 1
},
{
"definition": "continuous Hahn polynomial",
"score": 1
},
{
"definition": "continuous dual Hahn polynomial",
"score": 2
}
]
}
]
} |
| Gold 73 | P_n^{(\alpha,\beta)}=\lim_{t\to\infty}t^{-n}p_n\left(\tfrac12xt; \tfrac12(\alpha+1-it), \tfrac12(\beta+1+it), \tfrac12(\alpha+1+it), \tfrac12(\beta+1-it)\right)
|
Continuous Hahn polynomials |
|
|
- |
|
- |
|
- | Hidden argument cause mismatch. | Full data:
{
"id": 73,
"pid": 125,
"eid": "math.125.15",
"title": "Continuous Hahn polynomials",
"formulae": [
{
"id": "FORMULA_ff971744100fef3b34b2c93b6adc3efb",
"formula": "P_n^{(\\alpha,\\beta)}=\\lim_{t\\to\\infty}t^{-n}p_n\\left(\\tfrac12xt; \\tfrac12(\\alpha+1-it), \\tfrac12(\\beta+1+it), \\tfrac12(\\alpha+1+it), \\tfrac12(\\beta+1-it)\\right)",
"semanticFormula": "\\JacobipolyP{\\alpha}{\\beta}{n}@{x} = \\lim_{t\\to\\infty} t^{-n} \\contHahnpolyp{n}@{\\tfrac12 xt}{\\tfrac12(\\alpha + 1 - \\iunit t)}{\\tfrac12(\\beta + 1 + \\iunit t)}{\\tfrac12(\\alpha + 1 + \\iunit t)}{\\tfrac12(\\beta + 1 - \\iunit t)}",
"confidence": 0.9041995034970904,
"translations": {
"Mathematica": {
"translation": "JacobiP[n, \\[Alpha], \\[Beta], x] == Limit[(t)^(- n)* I^(n)*Divide[Pochhammer[Divide[1,2]*(\\[Alpha]+ 1 - I*t) + Divide[1,2]*(\\[Alpha]+ 1 + I*t), n]*Pochhammer[Divide[1,2]*(\\[Alpha]+ 1 - I*t) + Divide[1,2]*(\\[Beta]+ 1 - I*t), n], (n)!] * HypergeometricPFQ[{-(n), n + 2*Re[Divide[1,2]*(\\[Alpha]+ 1 - I*t) + Divide[1,2]*(\\[Beta]+ 1 + I*t)] - 1, Divide[1,2]*(\\[Alpha]+ 1 - I*t) + I*(Divide[1,2]*x*t)}, {Divide[1,2]*(\\[Alpha]+ 1 - I*t) + Divide[1,2]*(\\[Alpha]+ 1 + I*t), Divide[1,2]*(\\[Alpha]+ 1 - I*t) + Divide[1,2]*(\\[Beta]+ 1 - I*t)}, 1], t -> Infinity, GenerateConditions->None]"
}
},
"positions": [
{
"section": 5,
"sentence": 2,
"word": 20
}
],
"includes": [
"p_{n}(x;a,b,c,d)",
"F_{n}",
"P_{n}^{(\\alpha,\\beta)}"
],
"isPartOf": [],
"definiens": [
{
"definition": "case of the continuous Hahn polynomial",
"score": 1
},
{
"definition": "Jacobi polynomial",
"score": 2
},
{
"definition": "continuous Hahn polynomial",
"score": 2
}
]
}
]
} |
| Gold 74 | \sum^{b-1}_{s=a}w_n^{(c)}(s,a,b)w_m^{(c)}(s,a,b)\rho(s)[\Delta x(s-\frac{1}{2}) ]=\delta_{nm}d_n^2
|
Dual Hahn polynomials |
|
- | - |
|
- | - | - | Not standard notation for dual Hahn polynomial. DLMF uses R. Further, dual Hahn does not exist in Mathematica. | Full data:
{
"id": 74,
"pid": 126,
"eid": "math.126.7",
"title": "Dual Hahn polynomials",
"formulae": [
{
"id": "FORMULA_657ec9a2e460e61adc6857260291be56",
"formula": "\\sum^{b-1}_{s=a}w_n^{(c)}(s,a,b)w_m^{(c)}(s,a,b)\\rho(s)[\\Delta x(s-\\frac{1}{2}) ]=\\delta_{nm}d_n^2",
"semanticFormula": "\\sum_{s=a}^{b-1} \\dualHahnpolyR{n}@{c}{s}{a}{b} \\dualHahnpolyR{m}@{c}{s}{a}{b} \\rho(s) [\\Delta x(s - \\frac{1}{2})] = \\delta_{nm} d_n^2",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 8
}
],
"includes": [
"n"
],
"isPartOf": [],
"definiens": [
{
"definition": "Dual Hahn polynomial",
"score": 2
}
]
}
]
} |
| Gold 75 | p_n(x;a,b,c|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_n*_4\Phi_3(q^{-n},abcdq^{n-1},ae^{i{(t+2u)}},ae^{-it};abe^{2iu},ac,ad;q;q)
|
Continuous q-Hahn polynomials |
|
|
|
|
- | - | - | Asterisk has index. Wrong LaTeX from Wikipedia Editor. | Full data:
{
"id": 75,
"pid": 127,
"eid": "math.127.0",
"title": "Continuous q-Hahn polynomials",
"formulae": [
{
"id": "FORMULA_67e28846328978f4e08bb6b69fe6c549",
"formula": "p_n(x;a,b,c|q)=a^{-n}e^{-inu}(abe^{2iu},ac,ad;q)_n*_4\\Phi_3(q^{-n},abcdq^{n-1},ae^{i{(t+2u)}},ae^{-it};abe^{2iu},ac,ad;q;q)",
"semanticFormula": "p_n(x ; a , b , c|q) = a^{-n} \\expe^{-\\iunit nu} \\qmultiPochhammersym{ab\\expe^{2\\iunit u} , ac , ad}{q}{n} * \\qgenhyperphi{4}{3}@{q^{-n} , abcdq^{n-1} , a\\expe^{\\iunit{(t+2u)}} , a\\expe^{-\\iunit t}}{ab\\expe^{2\\iunit u} , ac , ad}{q}{q}",
"confidence": 0.8662724998444776,
"translations": {
"Mathematica": {
"translation": "p[n_, x_, a_, b_, c_, q_] := (a)^(- n)* Exp[- I*\\[Nu]]*Product[QPochhammer[Part[{a*b*Exp[2*I*u], a*c , a*d},i],q,n],{i,1,Length[{a*b*Exp[2*I*u], a*c , a*d}]}]* QHypergeometricPFQ[{(q)^(- n), a*b*c*d*(q)^(n - 1), a*Exp[I*(t + 2*u)], a*Exp[- I*t]},{a*b*Exp[2*I*u], a*c , a*d},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 1
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "continuous FORMULA_7694f4a66316e53c8cdd9d9954bd611d - Hahn polynomial",
"score": 2
},
{
"definition": "q - Pochhammer symbol",
"score": 2
}
]
}
]
} |
| Gold 76 | p_n(x;a,b,c\mid q)=\frac{(ab,ac;q)_n}{a^n}\cdot {_3\Phi_2}(q^-n,ae^{i\theta},ae^{-i\theta}; ab, ac \mid q;q)
|
Continuous dual q-Hahn polynomials |
|
|
|
|
- | - | Wrong LaTeX. q^-n only puts into the subscript but not .
|
Underscore mismatch. | Full data:
{
"id": 76,
"pid": 128,
"eid": "math.128.0",
"title": "Continuous dual q-Hahn polynomials",
"formulae": [
{
"id": "FORMULA_95daf919f18506606090e49a38d1c1a6",
"formula": "p_n(x;a,b,c\\mid q)=\\frac{(ab,ac;q)_n}{a^n}\\cdot {_3\\Phi_2}(q^-n,ae^{i\\theta},ae^{-i\\theta}; ab, ac \\mid q;q)",
"semanticFormula": "p_n(x ; a , b , c \mid q) = \frac{\qmultiPochhammersym{ab , ac}{q}{n}}{a^n} \cdot \qgenhyperphi{3}{2}@{q^{- n} , a\expe^{\iunit \theta} , a\expe^{- \iunit \theta}}{ab , ac}{q}{q}",
"confidence": 0.8662724998444776,
"translations": {
"Mathematica": {
"translation": "p[n_, x_, a_, b_, c_, q_] := Divide[Product[QPochhammer[Part[{a*b , a*c},i],q,n],{i,1,Length[{a*b , a*c}]}],(a)^(n)] * QHypergeometricPFQ[{(q)^(- n), a*Exp[I*\[Theta]], a*Exp[- I*\[Theta]]},{a*b , a*c},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "continuous dual FORMULA_7694f4a66316e53c8cdd9d9954bd611d - Hahn polynomial",
"score": 2
}
]
}
]
} |
| Gold 77 | Q_n(x;a,b,N;q)=\;_{3}\phi_2\left[\begin{matrix} q^-n & abq^n+1 & x \\ aq & q^-N \end{matrix} ; q,q \right]
|
Q-Hahn polynomials |
|
|
|
- | - | - | - | Cannot detect name of function. | Full data:
{
"id": 77,
"pid": 129,
"eid": "math.129.0",
"title": "Q-Hahn polynomials",
"formulae": [
{
"id": "FORMULA_b3a9ac90714e1e705d2a88b30e79cca0",
"formula": "Q_n(x;a,b,N;q)=\\;_{3}\\phi_2\\left[\\begin{matrix} q^-n & abq^n+1 & x \\\\ aq & q^-N \\end{matrix} ; q,q \\right]",
"semanticFormula": "\\qHahnpolyQ{n}@{x}{a}{b}{N}{q} = \\qgenhyperphi{3}{2}@{q^-n , abq^n+1 , x}{aq , q^-N}{q}{q}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "Q[n_, x_, a_, b_, N_, q_] := QHypergeometricPFQ[{(q)^(-)* n , a*b*(q)^(n)+ 1 , x},{a*q , (q)^(-)* N},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "q - Hahn polynomial",
"score": 2
},
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 0
}
]
}
]
} |
| Gold 78 | x=
|
Al-Salam–Chihara polynomials |
|
|
- | - | - | - | - | Wrong math detection. | Full data:
{
"id": 78,
"pid": 131,
"eid": "math.131.0",
"title": "Al-Salam\u2013Chihara polynomials",
"formulae": [
{
"id": "FORMULA_52a07ce46212cbc2298415c5fca6e075",
"formula": "x=",
"semanticFormula": "x=\\cos@{\\theta}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "x = Cos[\\[Theta]]"
},
"Maple": {
"translation": "x = cos(theta)"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 20
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "cosine function",
"score": 2
},
{
"definition": "substitution",
"score": 2
}
]
}
]
} |
| Gold 79 | \Phi_n^*(z)=z^n\overline{\Phi_n(1/\overline{z})}
|
Orthogonal polynomials on the unit circle |
|
|
- |
|
- | - | - | Nested overline didnt match (bug). | Full data:
{
"id": 79,
"pid": 132,
"eid": "math.132.7",
"title": "Orthogonal polynomials on the unit circle",
"formulae": [
{
"id": "FORMULA_f2d41903301a99a3fade5f2f49450694",
"formula": "\\Phi_n^*(z)=z^n\\overline{\\Phi_n(1\/\\overline{z})}",
"semanticFormula": "\\Phi_n^*(z) = z^n{\\conj{\\Phi_n(1 \/ \\conj{z})}}",
"confidence": 0.7579553437219001,
"translations": {
"Mathematica": {
"translation": "\\[CapitalPhi]\\[Prima][n_, z_] := z^n*Conjugate[\\[CapitalPhi][n, Divide[1, Conjugate[z]]]]"
}
},
"positions": [
{
"section": 2,
"sentence": 0,
"word": 8
}
],
"includes": [
"\\Phi_n(z)",
"z^n",
"\\alpha_n"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 2
}
]
}
]
} |
| Gold 80 | P_n(x) = c_n \, \det \begin{bmatrix}m_0 & m_1 & m_2 &\cdots & m_n \\m_1 & m_2 & m_3 &\cdots & m_{n+1} \\&&\vdots&& \vdots \\m_{n-1} &m_n& m_{n+1} &\cdots &m_{2n-1}\\1 & x & x^2 & \cdots & x^n\end{bmatrix}
|
Orthogonal polynomials |
|
- | - | - | - | - | - | No direct translation possible (indef number of arguments). | Full data:
{
"id": 80,
"pid": 133,
"eid": "math.133.8",
"title": "Orthogonal polynomials",
"formulae": [
{
"id": "FORMULA_c0641714ec593f58211623652c4a34f0",
"formula": "P_n(x) = c_n \\, \\det \\begin{bmatrix}m_0 & m_1 & m_2 &\\cdots & m_n \\\\m_1 & m_2 & m_3 &\\cdots & m_{n+1} \\\\&&\\vdots&& \\vdots \\\\m_{n-1} &m_n& m_{n+1} &\\cdots &m_{2n-1}\\\\1 & x & x^2 & \\cdots & x^n\\end{bmatrix}",
"semanticFormula": "P_n(x) = c_n \\det \\begin{bmatrix}m_0 & m_1 & m_2 &\\cdots & m_n \\\\m_1 & m_2 & m_3 &\\cdots & m_{n+1} \\\\&&\\vdots&& \\vdots \\\\m_{n-1} &m_n& m_{n+1} &\\cdots &m_{2n-1}\\\\1 & x & x^2 & \\cdots & x^n\\end{bmatrix}",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 5,
"sentence": 0,
"word": 16
}
],
"includes": [
"P_{n}",
"c_{n}",
"P_{m}"
],
"isPartOf": [],
"definiens": [
{
"definition": "constant",
"score": 0
},
{
"definition": "normalisation",
"score": 0
},
{
"definition": "orthogonal polynomial",
"score": 2
},
{
"definition": "term of the moment",
"score": 0
}
]
}
]
} |
| Gold 81 | \displaystyle p_n(x;a,b;q) = {}_2\phi_1(q^{-n},abq^{n+1};aq;q,xq)
|
Little q-Jacobi polynomials |
|
|
|
- | - | - |
|
No translation for \littleJacobipolyp
|
Full data:
{
"id": 81,
"pid": 134,
"eid": "math.134.0",
"title": "Little q-Jacobi polynomials",
"formulae": [
{
"id": "FORMULA_c492265e4cd4beeeb776dad843dc1f73",
"formula": "\\displaystyle p_n(x;a,b;q) = {}_2\\phi_1(q^{-n},abq^{n+1};aq;q,xq)",
"semanticFormula": "\\littleqJacobipolyp{n}@{x}{a}{b}{q} = \\qgenhyperphi{2}{1}@{q^{-n} , abq^{n+1}}{aq}{q}{xq}",
"confidence": 0.7229065246531701,
"translations": {
"Mathematica": {
"translation": "p[n_, x_, a_, b_, q_] := QHypergeometricPFQ[{(q)^(- n), a*b*(q)^(n + 1)},{a*q},q,x*q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 19
}
],
"includes": [
"q",
"p_{n}(x;a,b;q)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Jacobi polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 0
},
{
"definition": "little q - Jacobi polynomial",
"score": 2
}
]
}
]
} |
| Gold 82 | \displaystyle P_n(x;a,b,c;q)={}_3\phi_2(q^{-n},abq^{n+1},x;aq,cq;q,q)
|
Big q-Jacobi polynomials |
|
|
|
- | - | - |
|
No translation for \bigqJacobipolyP
|
Full data:
{
"id": 82,
"pid": 135,
"eid": "math.135.0",
"title": "Big q-Jacobi polynomials",
"formulae": [
{
"id": "FORMULA_0680f701a101288f89487a7a3fabefb1",
"formula": "\\displaystyle P_n(x;a,b,c;q)={}_3\\phi_2(q^{-n},abq^{n+1},x;aq,cq;q,q)",
"semanticFormula": "\\bigqJacobipolyP{n}@{x}{a}{b}{c}{q} = \\qgenhyperphi{3}{2}@{q^{-n} , abq^{n+1} , x}{aq , cq}{q}{q}",
"confidence": 0.7424814142326033,
"translations": {
"Mathematica": {
"translation": "p[n_, x_, a_, b_, c_, q_] := QHypergeometricPFQ[{(q)^(- n), a*b*(q)^(n + 1), x},{a*q , c*q},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 11
}
],
"includes": [
"P_{n}(x;a,b,c;q)",
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "big q - Jacobi polynomial",
"score": 2
}
]
}
]
} |
| Gold 83 | P_n(x;a,b;q)=\frac{1}{(b^{-1}*q^{-n};q,n)}*_2\Phi_1(q^{-n},aqx^{-1};aq|q;\frac{x}{b})
|
Big q-Laguerre polynomials |
|
|
- |
|
- | - | - | Again, invalid latex. The asterisk has the underscore. | Full data:
{
"id": 83,
"pid": 137,
"eid": "math.137.0",
"title": "Big q-Laguerre polynomials",
"formulae": [
{
"id": "FORMULA_aa5a6972c7e8327e316eddc8fd8e9b08",
"formula": "P_n(x;a,b;q)=\\frac{1}{(b^{-1}*q^{-n};q,n)}*_2\\Phi_1(q^{-n},aqx^{-1};aq|q;\\frac{x}{b})",
"semanticFormula": "P_n(x;a,b;q) =\\frac{1}{\\qmultiPochhammersym{b^{-1}*q^{-n}}{q}{n}} * \\qgenhyperphi{2}{1}@{q^{-n},aqx^{-1}}{aq}{q}{\\frac{x}{b}}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "P[n_, x_, a_, b_, q_] := Divide[1,Product[QPochhammer[Part[{(b)^(- 1)* (q)^(- n)},i],q,n],{i,1,Length[{(b)^(- 1)* (q)^(- n)}]}]]* QHypergeometricPFQ[{(q)^(- n), a*q*(x)^(- 1)},{a*q},q,Divide[x,b]]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 1
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "q - Pochhammer symbol",
"score": 1
},
{
"definition": "big q - Laguerre polynomial",
"score": 2
}
]
}
]
} |
| Gold 84 | K_n(\lambda(x);c,N|q)=_3\Phi_2(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q)
|
Dual q-Krawtchouk polynomials |
|
- | - |
|
- | - | - | Illegal LaTeX. Equal sign has underscore 3 (which is wrong). Further, dual q-Krawtchouk do not exist in Mathematica. | Full data:
{
"id": 84,
"pid": 138,
"eid": "math.138.0",
"title": "Dual q-Krawtchouk polynomials",
"formulae": [
{
"id": "FORMULA_9221dfda453868628eb8bbcd2d414fdf",
"formula": "K_n(\\lambda(x);c,N|q)=_3\\Phi_2(q^{-n},q^{-x},cq^{x-N};q^{-N},0|q;q)",
"semanticFormula": "K_n(\\lambda(x);c,N|q) = \\qgenhyperphi{3}{2}@{q^{-n},q^{-x},cq^{x-N}}{q^{-N},0}{q}{q}",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 0
},
{
"definition": "dual q - Krawtchouk polynomial",
"score": 2
}
]
}
]
} |
| Gold 85 | P_{n}^{(\alpha)}(x|q)=\frac{(q^\alpha+1;q)_{n}}{(q;q)_{n}}
|
Continuous q-Laguerre polynomials |
|
|
|
|
- | - | - | Did not detect q-multi Pochhammer symbol. | Full data:
{
"id": 85,
"pid": 139,
"eid": "math.139.0",
"title": "Continuous q-Laguerre polynomials",
"formulae": [
{
"id": "FORMULA_8c9e3af3c57272f3a6ddabba68ab4d3e",
"formula": "P_{n}^{(\\alpha)}(x|q)=\\frac{(q^\\alpha+1;q)_{n}}{(q;q)_{n}}",
"semanticFormula": "P_{n}^{(\\alpha)}(x|q) = \\frac{\\qmultiPochhammersym{q^\\alpha+1}{q}{n}}{\\qPochhammer{q}{q}{n}} \\qgenhyperphi{3}{2}@{q^{-n},q^{\\alpha\/2+1\/4}\\expe^{\\iunit\\theta},q^{\\alpha\/2+1\/4}*\\expe^{-\\iunit\\theta}}{q^{\\alpha+1},0}{q}{q}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "P[n_, \\[Alpha]_, x_, q_] := Divide[Product[QPochhammer[Part[{(q)^\\[Alpha]+ 1},i],q,n],{i,1,Length[{(q)^\\[Alpha]+ 1}]}],QPochhammer[q, q, n]]*QHypergeometricPFQ[{(q)^(- n), (q)^(\\[Alpha]\/2 + 1\/4)* Exp[I*\\[Theta]], (q)^(\\[Alpha]\/2 + 1\/4)* Exp[- I*\\[Theta]]},{(q)^(\\[Alpha]+ 1), 0},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 1,
"word": 0
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "continuous q - Laguerre polynomial",
"score": 2
},
{
"definition": "family of basic hypergeometric orthogonal polynomial",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 2
}
]
}
]
} |
| Gold 86 | \displaystyle p_n(x;a|q) = {}_2\phi_1(q^{-n},0;aq;q,qx) = \frac{1}{(a^{-1}q^{-n};q)_n}{}_2\phi_0(q^{-n},x^{-1};;q,x/a)
|
Little q-Laguerre polynomials |
|
|
|
|
- | - | - | Could not match empty arguments (bug). | Full data:
{
"id": 86,
"pid": 142,
"eid": "math.142.0",
"title": "Little q-Laguerre polynomials",
"formulae": [
{
"id": "FORMULA_4e548bca196e13d5af0eaadf2ea725d1",
"formula": "\\displaystyle p_n(x;a|q) = {}_2\\phi_1(q^{-n},0;aq;q,qx) = \\frac{1}{(a^{-1}q^{-n};q)_n}{}_2\\phi_0(q^{-n},x^{-1};;q,x\/a)",
"semanticFormula": "p_n(x ; a|q) = \\qgenhyperphi{2}{1}@{q^{-n} , 0}{aq}{q}{qx} = \\frac{1}{\\qmultiPochhammersym{a^{-1} q^{-n}}{q}{n}} \\qgenhyperphi{2}{0}@{q^{-n} , x^{-1}}{}{q}{x\/a}",
"confidence": 0.7219509974881755,
"translations": {
"Mathematica": "p[n_, x_, a_, q_] := QHypergeometricPFQ[{(q)^(- n), 0},{a*q},q,q*x] == Divide[1,Product[QPochhammer[Part[{(a)^(- 1)* (q)^(- n)},i],q,n],{i,1,Length[{(a)^(- 1)* (q)^(- n)}]}]]*QHypergeometricPFQ[{(q)^(- n), (x)^(- 1)},{},q,x\/a]"
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 15
}
],
"includes": [
"q",
"p_{n}(x;a|q)"
],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "little q - Laguerre polynomial",
"score": 2
},
{
"definition": "q - Pochhammer symbol",
"score": 2
}
]
}
]
} |
| Gold 87 | y_{n}(x;a;q)=\;_{2}\phi_1 \left(\begin{matrix} q^{-N} & -aq^{n} \\ 0 \end{matrix} ; q,qx \right)
|
Q-Bessel polynomials |
|
|
|
|
- | - | - | Wrong LaTeX. Equal sign has subsript 2. | Full data:
{
"id": 87,
"pid": 143,
"eid": "math.143.0",
"title": "Q-Bessel polynomials",
"formulae": [
{
"id": "FORMULA_c89da2fda6f9f6411ed4292f6d845f52",
"formula": "y_{n}(x;a;q)=\\;_{2}\\phi_1 \\left(\\begin{matrix} q^{-N} & -aq^{n} \\\\ 0 \\end{matrix} ; q,qx \\right)",
"semanticFormula": "y_{n}(x;a;q) = \\qgenhyperphi{2}{1}@{q^{-N} , -aq^{n}}{0}{q}{qx}",
"confidence": 0.6264217257193126,
"translations": {
"Mathematica": "y[n_, x_, a_, q_] := QHypergeometricPFQ[{(q)^(- N), - a*(q)^(n)},{0},q,q*x]"
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 16
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 0
},
{
"definition": "q - Bessel polynomial",
"score": 1
}
]
}
]
} |
| Gold 88 | h_n(ix;q^{-1}) = i^n\hat h_n(x;q)
|
Discrete q-Hermite polynomials |
|
- | - |
|
- | - | - | We correctly identified \discqHermitepolyhI but were not able to distinguish it from discqHermitepolyhII from RHS. Neither of them is translatable though.
|
Full data:
{
"id": 88,
"pid": 144,
"eid": "math.144.2",
"title": "Discrete q-Hermite polynomials",
"formulae": [
{
"id": "FORMULA_b9974285610b7a82c94b6a504726df8c",
"formula": "h_n(ix;q^{-1}) = i^n\\hat h_n(x;q)",
"semanticFormula": "\\discqHermitepolyhI{n}@{\\iunit x}{q^{-1}} = \\iunit^n \\discqHermitepolyhII{n}@{x}{q}",
"confidence": 0.8429359579302446,
"translations": {},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 27
}
],
"includes": [
"\\hat{h}_{n}(x;q)",
"q",
"h_{n}(x;q)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Hermite polynomial",
"score": 2
},
{
"definition": "term of basic hypergeometric function",
"score": 1
},
{
"definition": "Carlitz polynomial",
"score": 1
},
{
"definition": "Al-Salam",
"score": 1
},
{
"definition": "discrete q - Hermite polynomial",
"score": 2
}
]
}
]
} |
| Gold 89 | P_{n}(x;a\mid q) = a^{-n} e^{in\phi} \frac{a^2;q_n}{(q;q)_n} {_3}\Phi_2(q^-n, ae^{i(\theta+2\phi)}, ae^{-i\theta}; a^2, 0 \mid q; q)
|
Q-Meixner–Pollaczek polynomials |
|
|
|
|
- | - | - | Did not match underscore {_3}
|
Full data:
{
"id": 89,
"pid": 145,
"eid": "math.145.0",
"title": "Q-Meixner\u2013Pollaczek polynomials",
"formulae": [
{
"id": "FORMULA_fa6650cad7aed4d975716018ef03068f",
"formula": "P_{n}(x;a\\mid q) = a^{-n} e^{in\\phi} \\frac{a^2;q_n}{(q;q)_n} {_3}\\Phi_2(q^-n, ae^{i(\\theta+2\\phi)}, ae^{-i\\theta}; a^2, 0 \\mid q; q)",
"semanticFormula": "P_{n}(x ; a \\mid q) = a^{-n} \\expe^{\\iunit n\\phi} \\frac{\\qmultiPochhammersym{a^2}{q}{n}}{\\qmultiPochhammersym{q}{q}{n}} \\qgenhyperphi{3}{2}@{q^- n , a\\expe^{\\iunit(\\theta + 2 \\phi)} , a\\expe^{- \\iunit \\theta}}{a^2, 0}{q}{q}",
"confidence": 0.8662724998444776,
"translations": {
"Mathematica": {
"translation": "P[n_, x_, a_, q_] := (a)^(- n)* Exp[I*n*\\[Phi]]*Divide[Product[QPochhammer[Part[{(a)^(2)},i],q,n],{i,1,Length[{(a)^(2)}]}],Product[QPochhammer[Part[{q},i],q,n],{i,1,Length[{q}]}]]*QHypergeometricPFQ[{(q)^(-)* n , a*Exp[I*(\\[Theta]+ 2*\\[Phi])], a*Exp[- I*\\[Theta]]},{(a)^(2), 0},q,q]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 16
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "polynomial",
"score": 1
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "q - Pochhammer symbol",
"score": 2
},
{
"definition": "Q Meixner \u2013 Pollaczek polynomials",
"score": 2
}
]
}
]
} |
| Gold 90 | \displaystyle L_n^{(\alpha)}(x;q) = \frac{(q^{\alpha+1};q)_n}{(q;q)_n} {}_1\phi_1(q^{-n};q^{\alpha+1};q,-q^{n+\alpha+1}x)
|
Q-Laguerre polynomials |
|
|
|
- | - | - |
|
- | Full data:
{
"id": 90,
"pid": 149,
"eid": "math.149.0",
"title": "Q-Laguerre polynomials",
"formulae": [
{
"id": "FORMULA_dea0af895f73964b98741e71bc0635cb",
"formula": "\\displaystyle L_n^{(\\alpha)}(x;q) = \\frac{(q^{\\alpha+1};q)_n}{(q;q)_n} {}_1\\phi_1(q^{-n};q^{\\alpha+1};q,-q^{n+\\alpha+1}x)",
"semanticFormula": "\\qLaguerrepolyL{\\alpha}{n}@{x}{q} = \\frac{\\qmultiPochhammersym{q^{\\alpha+1}}{q}{n}}{\\qmultiPochhammersym{q}{q}{n}} \\qgenhyperphi{1}{1}@{q^{-n}}{q^{\\alpha+1}}{q}{- q^{n+\\alpha+1} x}",
"confidence": 0.779734956061429,
"translations": {
"Mathematica": {
"translation": "L[n_, \\[Alpha]_, x_, q_] := Divide[Product[QPochhammer[Part[{(q)^(\\[Alpha]+ 1)},i],q,n],{i,1,Length[{(q)^(\\[Alpha]+ 1)}]}],Product[QPochhammer[Part[{q},i],q,n],{i,1,Length[{q}]}]]*QHypergeometricPFQ[{(q)^(- n)},{(q)^(\\[Alpha]+ 1)},q,- (q)^(n + \\[Alpha]+ 1)* x]"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 18
}
],
"includes": [
"q",
"P_{n}^{(\\alpha)}(x;q)"
],
"isPartOf": [],
"definiens": [
{
"definition": "Laguerre polynomial",
"score": 2
},
{
"definition": "q - Laguerre polynomial",
"score": 2
},
{
"definition": "term of basic hypergeometric function",
"score": 2
},
{
"definition": "Pochhammer symbol",
"score": 1
},
{
"definition": "q - Pochhammer symbol",
"score": 2
}
]
}
]
} |
| Gold 91 | \sum_{n=0}^\infty H_n(x \mid q) \frac{t^n}{(q;q)_n} = \frac{1}{\left( t e^{i \theta},t e^{-i \theta};q \right)_\infty}
|
Continuous q-Hermite polynomials |
|
- | - |
|
- | - | - | Mistakenly detect Hermite polynomial but was continuous q-Hermite polynomial. | Full data:
{
"id": 91,
"pid": 150,
"eid": "math.150.3",
"title": "Continuous q-Hermite polynomials",
"formulae": [
{
"id": "FORMULA_a10dc9de9b2b618ad2f2e96dc9eb0207",
"formula": "\\sum_{n=0}^\\infty H_n(x \\mid q) \\frac{t^n}{(q;q)_n} = \\frac{1}{\\left( t e^{i \\theta},t e^{-i \\theta};q \\right)_\\infty}",
"semanticFormula": "\\sum_{n=0}^\\infty \\contqHermitepolyH{n}@{x}{q} \\frac{t^n}{\\qmultiPochhammersym{q}{q}{n}} = \\frac{1}{\\qmultiPochhammersym{t \\expe^{\\iunit \\theta} , t \\expe^{- \\iunit \\theta}}{q}{\\infty}}",
"confidence": 0.7796357038819148,
"translations": {},
"positions": [
{
"section": 3,
"sentence": 0,
"word": 0
}
],
"includes": [
"q"
],
"isPartOf": [],
"definiens": [
{
"definition": "continuous q - Hermite polynomial",
"score": 2
},
{
"definition": "q - Pochhammer symbol",
"score": 2
}
]
}
]
} |
| Gold 92 | w^{\prime\prime}+\xi\sin(2z)w^{\prime}+(\eta-p\xi\cos(2z))w=0.
|
Ince equation |
|
|
- | - |
|
|
- | ODE. | Full data:
{
"id": 92,
"pid": 151,
"eid": "math.151.0",
"title": "Ince equation",
"formulae": [
{
"id": "FORMULA_ce9ed9f979f486263028e3d86b63ac60",
"formula": "w^{\\prime\\prime}+\\xi\\sin(2z)w^{\\prime}+(\\eta-p\\xi\\cos(2z))w=0. ",
"semanticFormula": "w^{\\prime\\prime}+\\xi\\sin(2z)w^{\\prime}+(\\eta-p\\xi\\cos(2z))w=0",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "D[w[z], {z, 2}] + \\[Xi]*Sin[2*z]*D[w[z], {z, 1}] + (\\[Eta]-p*\\[Xi]*Cos[2*z])*w[z] == 0"
}
},
"positions": [
{
"section": 0,
"sentence": 0,
"word": 19
}
],
"includes": [
"p"
],
"isPartOf": [],
"definiens": [
{
"definition": "differential equation",
"score": 2
},
{
"definition": "Ince equation",
"score": 2
},
{
"definition": "mathematics",
"score": 0
},
{
"definition": "non-negative integer",
"score": 0
},
{
"definition": "Edward Lindsay Ince",
"score": 0
},
{
"definition": "polynomial solution",
"score": 0
},
{
"definition": "Ince polynomial",
"score": 1
}
]
}
]
} |
| Gold 93 | Q_v^\mu(x)= \cos(\mu\pi)\left(\frac{1+x}{1-x}\right)^{\mu/2}\frac{F(v+1,-v;1-\mu;1/2-2/x)} {\Gamma(1-\mu ) }
|
Ferrers function |
|
|
- |
|
- |
|
- | No information about gamma fuction and hypergeometric function. | Full data:
{
"id": 93,
"pid": 152,
"eid": "math.152.1",
"title": "Ferrers function",
"formulae": [
{
"id": "FORMULA_b5ab87b9cd2da05be00884345889d9e3",
"formula": "Q_v^\\mu(x)= \\cos(\\mu\\pi)\\left(\\frac{1+x}{1-x}\\right)^{\\mu\/2}\\frac{F(v+1,-v;1-\\mu;1\/2-2\/x)} {\\Gamma(1-\\mu ) }",
"semanticFormula": "\\FerrersQ[\\mu]{v}@{x} = \\cos(\\mu \\cpi)(\\frac{1+x}{1-x})^{\\mu\/2} \\frac{\\hyperF@{v+1}{-v}{1-\\mu}{1\/2-2\/x}}{\\EulerGamma@{1-\\mu}}",
"confidence": 0.8133162393162393,
"translations": {
"Mathematica": {
"translation": "LegendreQ[v, \\[Mu], x] == Cos[(\\[Mu]*Pi)*]*(Divide[1 + x,1 - x])^(\\[Mu]\/2)*Divide[Hypergeometric2F1[v + 1, - v, 1 - \\[Mu], 1\/2 - 2\/x],Gamma[1 - \\[Mu]]]"
},
"Maple": {
"translation": "LegendreQ(v, mu, x) = cos((mu*Pi)*)*((1 + x)\/(1 - x))^(mu\/2)*(hypergeom([v + 1, - v], [1 - mu], 1\/2 - 2\/x))\/(GAMMA(1 - mu))"
}
},
"positions": [
{
"section": 1,
"sentence": 0,
"word": 13
}
],
"includes": [],
"isPartOf": [],
"definiens": [
{
"definition": "Ferrers function of the second kind",
"score": 2
},
{
"definition": "Ferrers function of the first kind",
"score": 1
},
{
"definition": "Gamma function",
"score": 2
},
{
"definition": "hypergeometric function",
"score": 2
}
]
}
]
} |
| Gold 94 | H_{-v}^{(1)}(z,w)=e^{v\pi i}H_v^{(1)}(z,w)
|
Incomplete Bessel functions |
|
- | - | - | - | - | - |
|
Full data:
{
"id": 94,
"pid": 153,
"eid": "math.153.27",
"title": "Incomplete Bessel functions",
"formulae": [
{
"id": "FORMULA_35ab66efafff0de40d98c0778ebb63c3",
"formula": "H_{-v}^{(1)}(z,w)=e^{v\\pi i}H_v^{(1)}(z,w)",
"semanticFormula": "H_{-v}^{(1)}(z,w) = \\expe^{v \\cpi \\iunit} H_v^{(1)}(z , w)",
"confidence": 0,
"translations": {},
"positions": [
{
"section": 2,
"sentence": 0,
"word": 16
}
],
"includes": [
"v",
"w",
"H_v^{(1)}(z,w)"
],
"isPartOf": [],
"definiens": [
{
"definition": "incomplete Bessel function",
"score": 2
}
]
}
]
} |
| Gold 95 | K_v(x,y)=\int_1^\infty\frac{e^{-xt-\frac{y}{t}}}{t^{v+1}}dt
|
Incomplete Bessel K function/generalized incomplete gamma function |
|
|
|
- | - | - | - | - | Full data:
{
"id": 95,
"pid": 154,
"eid": "math.154.0",
"title": "Incomplete Bessel K function\/generalized incomplete gamma function",
"formulae": [
{
"id": "FORMULA_c333a7966510ed0b8f4de3147eabe47a",
"formula": "K_v(x,y)=\\int_1^\\infty\\frac{e^{-xt-\\frac{y}{t}}}{t^{v+1}}dt",
"semanticFormula": "K_v(x , y) = \\int_1^\\infty \\frac{\\expe^{-xt-\\frac{y}{t}}}{t^{v+1}} \\diff{t}",
"confidence": 0,
"translations": {
"Mathematica": {
"translation": "K[v_, x_, y_] := Integrate[Divide[Exp[- x*t -Divide[y,t]],(t)^(v + 1)], {t, 1, Infinity}, GenerateConditions->None]"
}
},
"positions": [
{
"section": 0,
"sentence": 0,
"word": 18
}
],
"includes": [
"K_v(x,y)"
],
"isPartOf": [],
"definiens": [
{
"definition": "mathematician",
"score": 0
},
{
"definition": "type incomplete-version of Bessel function",
"score": 2
},
{
"definition": "type generalized-version of incomplete gamma function",
"score": 0
}
]
}
]
} |