MediaWiki:Gold-data.json

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id1
pid51
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}"
confidence0.8803349492974287
translations
Mathematica
translation"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]"
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
Empty array
tokenTranslations
\pgcd"Greatest common divisor; Example: \pgcd{m,n} Will be translated to: GCD[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/27.1#p2.t1.r3 Mathematica: https://reference.wolfram.com/language/ref/GCD.html"
\BesselY"Bessel function second kind; Example: \BesselY{v}@{z} Will be translated to: BesselY[$0, $1] Branch Cuts: (-\infty, 0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/10.2#E3 Mathematica: https://"
\BesselJ"Bessel function first kind; Example: \BesselJ{v}@{z} Will be translated to: BesselJ[$0, $1] Branch Cuts: if v \notin \Integers: (-\infty, 0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/10.2#E2 Mathematica: 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
Empty array
tokenTranslations
\pgcd"Greatest common divisor; Example: \pgcd{m,n} Will be translated to: gcd($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/27.1#p2.t1.r3 Maple: https://www.maplesoft.com/support/help/Maple/view.aspx?path=gcd"
\BesselY"Bessel function second kind; Example: \BesselY{v}@{z} Will be translated to: BesselY($0, $1) Branch Cuts: (-\infty, 0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/10.2#E3 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Bessel"
\BesselJ"Bessel function first kind; Example: \BesselJ{v}@{z} Will be translated to: BesselJ($0, $1) Branch Cuts: if v \notin \Integers: (-\infty, 0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/10.2#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Bessel"
positions
section8
sentence8
word32
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
Empty array
definiens
definition"Bessel function first kind"
score2
definition"Bessel function second kind"
score2
definition"above relation"
score0
definition"spherical Bessel"
score1
definition"integer"
score1
definition"nonnegative integer"
score1
definition"relationship"
score0
definition"function"
score1
definition"recurrence relation"
score1
definition"Bessel"
score1
definition"large number of other known integral"
score0
definition"positive zero"
score0
definition"entire function of genus"
score0
definition"identity"
score0
definition"orthogonality relation"
score0
definition"Bessel function"
score2
definition"term"
score0
definition"real zero"
score0
definition"similar relation"
score0
definition"Hankel"
score1
definition"Bessel function of the second kind"
score2
definition"limit"
score0
definition"ordinary Bessel function"
score1
definition"case"
score0
definition"negative integer"
score0
definition"integral formula"
score0
definition"small argument"
score0
definition"average"
score0
definition"Bessel function of the first kind"
score2
definition"reference"
score0
definition"series expansion"
score0
definition"spherical Bessel function"
score1
definition"Abel 's identity"
score0
definition"important property of Bessel 's equation"
score1
definition"particular Bessel"
score1
definition"solution of Bessel 's equation"
score0
definition"Wronskian of the solution"
score0
definition"series"
score0
definition"closure equation"
score0
id2
pid52
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}"
confidence0.8896531556938116
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
Empty array
constraints
Empty array
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} Will be translated to: EllipticE[($0)^2] Relevant links to definitions: DLMF: http://dlmf.nist.gov/19.2#E8 Mathematica: https://"
\sin"Sine; Example: \sin@@{z} Will be translated to: Sin[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E1 Mathematica: 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
Empty array
constraints
Empty array
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} Will be translated to: EllipticE($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/19.2#E8 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=EllipticE"
\sin"Sine; Example: \sin@@{z} Will be translated to: sin($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sin"
positions
section37
sentence0
word39
includes
"\theta"
"E"
"\pi a b"
"\pi"
"e"
"E(e)"
isPartOf
Empty array
definiens
definition"complete elliptic integral of the second kind"
score2
definition"elementary function"
score1
definition"function"
score1
definition"length of the semi-major axis"
score2
definition"eccentricity"
score2
definition"circumference"
score0
definition"ellipse"
score1
definition"angle"
score1
definition"angular coordinate"
score1
definition"center"
score0
definition"formula"
score0
definition"rotation angle"
score0
id3
pid53
eid"math.53.6"
title"Elliptic integral"
formulae
id"FORMULA_04e9de23897a3b23dee1a9b7312ad99e"
formula"F(x;k) = u"
semanticFormula"\incellintFk@{\asin@{\Jacobiellsnk@@{u}{k}}}{k} = u"
confidence0
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
Empty array
tokenTranslations
Empty object
Maple
translation"EllipticF(JacobiSN(u, k), k) = u"
translationInformation
subEquations
"EllipticF(JacobiSN(u, k), k) = u"
freeVariables
"k"
"u"
constraints
Empty array
tokenTranslations
Empty object
positions
section2
sentence6
word5
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"
score1
definition"Jacobian elliptic function"
score2
definition"Legendre"
score1
definition"normal form"
score1
definition"trigonometric form"
score1
definition"incomplete elliptic integral of the second kind"
score0
definition"incomplete elliptic integral of the first kind"
score2
id4
pid54
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}"
confidence0.8809245132365588
translations
Empty object
positions
section11
sentence10
word9
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
Empty array
definiens
definition"related expression"
score0
definition"integer"
score0
definition"reflection formula"
score1
definition"end"
score0
definition"Hankel contour"
score2
definition"Riemann sphere"
score1
definition"Hankel 's formula for the gamma function"
score2
definition"gamma function"
score2
definition"reciprocal gamma function"
score2
id5
pid55
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"
confidence0
translations
Mathematica
translation"(2)^(4) == 2 * 2 * 2 * 2 == 16"
translationInformation
subEquations
"(2)^(4) = 2 * 2 * 2 * 2"
"2 * 2 * 2 * 2 = 16"
freeVariables
Empty array
constraints
Empty array
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
Empty array
constraints
Empty array
tokenTranslations
\times"was translated to: *"
positions
section4
sentence0
word3
includes
"2"
"^{4}"
isPartOf
Empty array
definiens
definition"example"
score2
id6
pid56
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}"
confidence0.9073333333333333
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
section4
sentence7
word23
includes
"1"
"n"
"2"
"x"
"\psi"
isPartOf
Empty array
definiens
definition"analytic continuation"
score0
definition"absolute convergence"
score0
definition"convenience"
score0
definition"inversion"
score0
definition"process"
score0
definition"stricter requirement"
score0
definition"series"
score1
definition"definition"
score2
id7
pid57
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})"
confidence0.8728566391293461
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
Empty array
tokenTranslations
\logint"Logarithmic integral; Example: \logint@{x} Will be translated to: LogIntegral[$0] Constraints: x > 1 Mathematica uses other branch cuts: (-\inf, 1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/6.2#E8 Mathematica: https://reference.wolfram.com/language/ref/LogIntegral.html"
\ln"Natural logarithm; Example: \ln@@{z} Will be translated to: Log[$0] Constraints: z != 0 Branch Cuts: (-\infty, 0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.2#E2 Mathematica: 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
Empty array
tokenTranslations
\logint"Logarithmic integral; Example: \logint@{x} Will be translated to: Li($0) Constraints: x > 1 Relevant links to definitions: DLMF: http://dlmf.nist.gov/6.2#E8 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Li"
\ln"Natural logarithm; Example: \ln@@{z} Will be translated to: ln($0) Constraints: z != 0 Branch Cuts: (-\infty, 0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.2#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=ln"
positions
section1
sentence2
word22
includes
"x"
"x)"
isPartOf
Empty array
definiens
definition"Cauchy principal value"
score2
definition"function"
score1
definition"singularity"
score1
definition"special function"
score1
definition"integral representation"
score1
definition"integral logarithm li"
score2
definition"logarithmic integral function"
score2
definition"logarithmic integral"
score2
definition"function li"
score1
id8
pid58
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}"
confidence0
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
section5
sentence4
word24
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
Empty array
definiens
definition"yield"
score0
definition"integral expression for the weight"
score2
definition"integrand"
score1
definition"L'Hôpital 's rule"
score0
definition"limit"
score0
definition"polynomial of degree"
score0
id9
pid59
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}"
confidence0
translations
Mathematica
translation"x == ProductLog[x]*(E)^(ProductLog[x]) D[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
Empty array
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
Empty array
tokenTranslations
\expe"Recognizes e with power as the exponential function. It was translated as a function."
positions
section12
sentence1
word14
includes
"e^{w}"
"x"
isPartOf
Empty array
definiens
definition"substitution"
score2
definition"third identity"
score0
definition"second identity"
score1
id10
pid60
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}"
confidence0.808438593520797
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
section6
sentence0
word21
includes
"P_n(x)"
"P_n"
"P_n(\cos\theta)"
"P_{n}(x)"
"P_m"
"r"
"r{'}"
"\mathbf{x}"
"\mathbf{x}{'}"
"\gamma"
"P"
isPartOf
Empty array
definiens
definition"expansion"
score2
definition"Adrien-Marie Legendre as the coefficient"
score0
definition"angle"
score1
definition"Legendre polynomial"
score2
definition"length of the vector"
score1
definition"vector"
score1
definition"polynomial"
score1
id11
pid61
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"
confidence0.82607945540953
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} Will be translated to: D[$1, {$2, $0}] Relevant links to definitions: DLMF: http://dlmf.nist.gov/1.4#E4 Mathematica: https://"
\cpi"Pi was translated to: Pi"
\HermitepolyH"Hermite polynomial; Example: \HermitepolyH{n}@{x} Will be translated to: HermiteH[$0, $1] Relevant links to definitions: DLMF: http://dlmf.nist.gov/18.3#T1.t1.r13 Mathematica: https://"
\expe"Recognizes e with power as the exponential function. It was translated as a function."
\erf"Error function; Example: \erf@@{z} Will be translated to: Erf[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/7.2#E1 Mathematica: https://reference.wolfram.com/language/ref/Erf.html"
Maple
translation"diff(erf(z), [z$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} Will be translated to: diff($1, [$2$($0)]) Relevant links to definitions: DLMF: http://dlmf.nist.gov/1.4#E4 Maple: https://www.maplesoft.com/support/help/Maple/view.aspx?path=diff"
\cpi"Pi was translated to: Pi"
\HermitepolyH"Hermite polynomial; Example: \HermitepolyH{n}@{x} Will be translated to: HermiteH($0, $1) Relevant links to definitions: DLMF: http://dlmf.nist.gov/18.3#T1.t1.r13 Maple: 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} Will be translated to: erf($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/7.2#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=erf"
positions
section5
sentence4
word6
includes
"erf"
"e^{-t^2}"
"-1"
"z"
"z)"
"e"
"\mathit{H}"
"z^{\bar{n}}"
isPartOf
Empty array
definiens
definition"Higher order derivative"
score2
definition"physicists ' Hermite polynomial"
score1
definition"name error function"
score1
definition"erfc"
score1
definition"error function"
score2
definition"erf"
score1
id12
pid62
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"
confidence0
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} Will be translated to: Cos[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E2 Mathematica: 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} Will be translated to: cos($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E2 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=cos"
\cpi"Pi was translated to: Pi"
positions
section8
sentence2
word18
includes
"n"
"x"
"n x"
"-1"
"k = 0"
"x_{k}"
isPartOf
Empty array
definiens
definition"root"
score2
definition"one"
score0
definition"trigonometric definition"
score0
definition"fact"
score0
definition"different simple root"
score1
definition"Chebyshev polynomial of the first kind"
score1
definition"Chebyshev polynomial"
score1
id13
pid63
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})"
confidence0
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
section25
sentence2
word16
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
Empty array
definiens
definition"distributional identity"
score1
definition"separable kernel"
score1
definition"Mehler 's formula"
score2
definition"Hermite polynomial"
score1
definition"Hermite function"
score2
definition"Hermite"
score1
definition"bivariate Gaussian probability density"
score1
definition"Gaussian probability density"
score1
definition"Gaussian probability"
score1
id14
pid64
eid"math.64.8"
title"Legendre function"
formulae
id"FORMULA_06f9b7b1d3f141742ad1c582b55056ba"
formula"x = \pm 1"
semanticFormula"x = \pm 1"
confidence0
translations
Mathematica
translation"x == \[PlusMinus]1"
translationInformation
subEquations
"x = + 1"
"x = - 1"
freeVariables
"x"
constraints
Empty array
tokenTranslations
\pm"was translated to: \[PlusMinus]"
Maple
translation"x = &+- 1"
translationInformation
subEquations
"x = + 1"
"x = - 1"
freeVariables
"x"
constraints
Empty array
tokenTranslations
\pm"was translated to: &+-"
positions
section3
sentence1
word11
includes
Empty array
isPartOf
Empty array
definiens
definition"value"
score2
id15
pid65
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}}"
confidence0.8953028732079359
translations
Mathematica
translation"EulerE[n] == (2)^(n)* EulerE[n, Divide[1,2]]"
Maple
translation"euler(n) = (2)^(n)* euler(n, (1)/(2))"
positions
section8
sentence4
word6
includes
"B_{n}"
"n"
"E_{k}"
isPartOf
Empty array
definiens
definition"Euler number"
score2
id16
pid66
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}"
confidence0.8811682126384021
translations
Mathematica
translation"SinIntegral[I*x] == I*SinhIntegral[x]"
translationInformation
subEquations
"SinIntegral[I*x] == I*SinhIntegral[x]"
freeVariables
"x"
constraints
Empty array
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} Will be translated to: SinIntegral[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/6.2#E9 Mathematica: 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
Empty array
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} Will be translated to: Si($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/6.2#E9 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Si"
positions
section3
sentence1
word9
includes
"Si"
"Si(x)"
"x"
isPartOf
Empty array
definiens
definition"ordinary sine"
score1
definition"Trigonometric integral"
score2
definition"hyperbolic sine integral"
score2
id17
pid67
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}}"
confidence0.8953028732079359
translations
Mathematica
translation"f[x_, y_] := Divide[1,Beta[x, y]]"
Maple
translation"f := (x,y) -> (1)/(Beta(x, y))"
positions
section6
sentence0
word12
includes
"x, y"
"\Beta"
"y"
"x"
isPartOf
Empty array
definiens
definition"function about the form"
score0
definition"reciprocal beta function"
score2
definition"definite integral of trigonometric function"
score1
definition"integral representation"
score0
definition"product"
score0
definition"power"
score0
definition"multiple-angle"
score0
definition"beta function"
score2
id18
pid68
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}"
confidence0.869061849326977
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
section5
sentence0
word14
includes
"dx"
"x"
isPartOf
Empty array
definiens
definition"incomplete gamma function"
score2
definition"confluent hypergeometric function"
score2
definition"Fresnel integral"
score1
definition"imaginary part"
score1
id19
pid69
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})"
confidence0
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
section18
sentence3
word4
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
Empty array
definiens
definition"Rodrigues ' formula"
score2
definition"orthogonal polynomial"
score1
definition"Chebyshev polynomials of the second kind"
score1
definition"classical orthogonal polynomial"
score1
definition"Chebyshev polynomial"
score2
definition"Gamma function"
score2
id20
pid70
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}"
confidence0
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
section17
sentence2
word0
includes
"z"
"n"
"z)"
"_{p}F_{q}"
"^{n}"
isPartOf
Empty array
definiens
definition"geometric series with ratio"
score2
definition"coefficient"
score0
definition"hypergeometric function"
score2
id21
pid71
eid"math.71.1-1"
title"Dirichlet L-function"
formulae
id"FORMULA_dcb9beab8f504cfc907c3165d24e5ad3"
formula"\chi(-1) = 1"
semanticFormula"\Dirichletchar@@{- 1}{k} = 1"
confidence0.746792096089683
translations
Mathematica
translation"DirichletCharacter[1, 1, -1] == 1"
positions
section1
sentence0
word7
includes
"\chi"
isPartOf
"a=\begin{cases}0;&\mbox{if }\chi(-1)=1, \\ 1;&\mbox{if }\chi(-1)=-1,\end{cases}"
definiens
definition"primitive character"
score2
definition"integer"
score1
definition"only zero"
score0
definition"Gamma function"
score0
definition"symbol"
score0
definition"functional equation"
score0
definition"Gauss sum"
score0
definition"Dirichlet character"
score2
id22
pid72
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}}}]"
confidence0.6525418663370697
translations
Empty object
positions
section3
sentence9
word9
includes
"z"
"z)"
"= 0"
isPartOf
Empty array
definiens
definition"z"
score0
definition"asymptotic formula for Ai"
score1
definition"Bi"
score1
definition"asymptotic behaviour of the Airy function"
score1
definition"Ai"
score1
definition"cosine"
score2
definition"definition of the Airy function"
score1
definition"Airy function"
score2
definition"Gamma function"
score2
id23
pid73
eid"math.73.41"
title"Dawson function"
formulae
id"FORMULA_f6b555bd8ce626d90119ab5eafdaeff2"
formula"F'(y)=1-2yF(y)"
semanticFormula"\DawsonsintF'@{y}=1-2y\DawsonsintF@{y}"
confidence0
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
section2
sentence9
word1
includes
"y"
isPartOf
Empty array
definiens
definition"polynomial"
score0
definition"Dawson function"
score2
id24
pid74
eid"math.74.0-1"
title"Hurwitz zeta function"
formulae
id"FORMULA_80a3608d4c2aae63f082861007c16c38"
formula"s\not =1"
semanticFormula"s \neq 1"
confidence0
translations
Mathematica
translation"s \[NotEqual] 1"
positions
section0
sentence2
word24
includes
"s"
"1"
"\not = 1"
isPartOf
Empty array
definiens
definition"value"
score2
id25
pid75
eid"math.75.6-1"
title"Theta function"
formulae
id"FORMULA_bfba6c35dbbcd8b89c6a29b1ffd6f517"
formula"q = e^{i\pi\tau}"
semanticFormula"q = \expe^{\iunit \cpi \tau}"
confidence0
translations
Mathematica
translation"q == Exp[I*Pi*\[Tau]]"
Maple
translation"q = exp(I*Pi*tau)"
positions
section2
sentence0
word57
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"
score2
definition"nome"
score2
id26
pid76
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}"
confidence0.6954186066124032
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
section16
sentence0
word15
includes
"k"
"^{2}"
isPartOf
Empty array
definiens
definition"derivative"
score2
definition"elliptic function"
score2
definition"basic Jacobi"
score0
definition"sn"
score2
definition"dn"
score2
definition"cn"
score2
id27
pid77
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}}"
confidence0.8121295595054496
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
section25
sentence1
word15
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
Empty array
definiens
definition"Fourier"
score1
definition"upper incomplete Gamma function"
score2
definition"lower incomplete Gamma function"
score2
id28
pid78
eid"math.78.0-1"
title"Polylogarithm"
formulae
id"FORMULA_e939f30d07578c2fb0d8cb5201db3c79"
formula"_{1}(z) ="
semanticFormula"\polylog{1}@{z} = -\ln@{1-z}"
confidence0
translations
Mathematica
translation"PolyLog[1, z] = -Log[1 - z]"
Maple
translation"polylog(1, z) = -ln(1 - z)"
positions
section0
sentence7
word11
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"
score2
definition"logarithm"
score2
definition"polylogarithm function"
score2
definition"dilogarithm"
score1
definition"trilogarithm"
score1
id29
pid79
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)"
confidence0
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
section1
sentence9
word16
includes
"\pi"
"\infty"
"sinc"
isPartOf
Empty array
definiens
definition"argument"
score0
definition"continuous Fourier"
score2
definition"rectangular function"
score2
definition"sinc"
score2
definition"ordinary frequency"
score1
id30
pid80
eid"math.80.26"
title"Exponential integral"
formulae
id"FORMULA_a9a738ef9d4e46360dd9b87b39c691bf"
formula"N=1"
semanticFormula"N=1"
confidence0
translations
Mathematica
translation"N == 1"
Maple
translation"N = 1"
positions
section4
sentence3
word30
includes
"N"
isPartOf
Empty array
definiens
definition"large value"
score2
definition"value"
score2
id31
pid81
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}}"
confidence0.8953028732079359
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
section15
sentence0
word10
includes
"\alpha"
"L_{n}^{(\alpha)}"
"L_n^{(\alpha)}(x)"
"n"
isPartOf
Empty array
definiens
definition"Hille formula"
score2
definition"Laguerre polynomial"
score2
definition"series on the left converge"
score0
definition"generalized Laguerre polynomial"
score2
definition"confluent hypergeometric function"
score2
id32
pid82
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)!}"
confidence0
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
section1
sentence7
word26
includes
"m"
"(-1)^{m}"
"- 1"
isPartOf
Empty array
definiens
definition"proportionality constant"
score2
id33
pid83
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}"
confidence0.7929614010341081
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
Empty array
tokenTranslations
\ScorerGi"Scorer function Gi; Example: \ScorerGi@{z} Will be translated to: ScorerGi[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/9.12#i Mathematica: https://"
\cpi"Pi was translated to: Pi"
\sin"Sine; Example: \sin@@{z} Will be translated to: Sin[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E1 Mathematica: 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
Empty array
tokenTranslations
\ScorerGi"Scorer function Gi; Example: \ScorerGi@{z} Will be translated to: Alternative translations: [AiryBi($0)*(int(AiryAi(t), t = ($0) .. infinity))+AiryAi($0)*(int(AiryBi(t), t = 0 .. ($0)))]Relevant links to definitions: DLMF: http://dlmf.nist.gov/9.12#i Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=Airy"
\cpi"Pi was translated to: Pi"
\sin"Sine; Example: \sin@@{z} Will be translated to: sin($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/4.14#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=sin"
positions
section0
sentence1
word15
includes
"x"
"x)"
isPartOf
Empty array
definiens
definition"Scorer 's function"
score2
definition"special function"
score1
id34
pid84
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}"
confidence0.8620216359266987
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
section6
sentence0
word20
includes
"w(z)]"
"z"
"V(x;\sigma,\gamma)"
"x"
"w(z)"
isPartOf
Empty array
definiens
definition"term of the Faddeeva function"
score2
definition"second derivative profile"
score2
definition"real part of the Faddeeva function"
score2
definition"Faddeeva function"
score2
definition"Voigt function"
score2
definition"Voigt profile"
score2
id35
pid85
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}}"
confidence0.8662724998444776
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
section6
sentence1
word23
includes
"a"
"\Phi(z,s,a)"
"z"
"s"
isPartOf
Empty array
definiens
definition"asymptotic expansion"
score2
definition"Pochhammer symbol"
score1
definition"Lerch transcendent"
score2
definition"polylogarithm"
score2
definition"polylogarithm function"
score2
definition"Pochhammer symbol"
score2
id36
pid86
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"
confidence0.912945064646862
translations
Mathematica
translation"Hypergeometric1F1[1, 2, z] == (Exp[z]- 1)/z 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
section10
sentence4
word0
includes
"M"
"U(a, b, z)"
"z"
"U(n,c,z)"
"\Phi(a, b, z)"
"M(n,b,z)"
"M(a, b, z)"
isPartOf
Empty array
definiens
definition"etc"
score0
definition"Kummer 's function of the first kind"
score2
definition"confluent hypergeometric function"
score1
definition"hypergeometric function"
score1
id37
pid87
eid"math.87.54"
title"Mathieu function"
formulae
id"FORMULA_f694135eafc20195a9d96ca3ce8af674"
formula"\sigma = \pm 1"
semanticFormula"\sigma = \pm 1"
confidence0
translations
Mathematica
translation"\[Sigma] == \[PlusMinus]1"
translationInformation
subEquations
"\[Sigma] = + 1"
"\[Sigma] = - 1"
freeVariables
"\[Sigma]"
constraints
Empty array
tokenTranslations
\pm"was translated to: \[PlusMinus]"
Maple
translation"sigma = &+- 1"
translationInformation
subEquations
"sigma = + 1"
"sigma = - 1"
freeVariables
"sigma"
constraints
Empty array
tokenTranslations
\pm"was translated to: &+-"
positions
section4
sentence1
word27
includes
Empty array
isPartOf
Empty array
definiens
definition"value"
score2
id38
pid88
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"
confidence0
translations
Mathematica
translation"D[f[z], {z, 2}] + (a*z^2 + b*z + c)*f[z] == 0"
positions
section0
sentence0
word19
includes
"z"
isPartOf
Empty array
definiens
definition"solution to the differential equation"
score2
definition"special function"
score1
definition"mathematics"
score0
definition"parabolic cylinder function"
score1
id39
pid89
eid"math.89.23"
title"Painlevé transcendents"
formulae
id"FORMULA_0a306ab913684a1ba3935715d3dd8ad8"
formula"c=\infty"
semanticFormula"c=\infty"
confidence0
translations
Mathematica
translation"c == Infinity"
Maple
translation"c = infinity"
positions
section9
sentence5
word23
includes
"c"
isPartOf
Empty array
definiens
definition"central charge of the Virasoro algebra"
score2
definition"combination of conformal block"
score1
definition"Painlevé VI equation"
score1
definition"two-dimensional conformal field theory"
score1
id40
pid90
eid"math.90.7"
title"Hypergeometric function"
formulae
id"FORMULA_aaffb0ad8dea17d68491d9fb6ebcfbe3"
formula"c = a + 1"
semanticFormula"c = a + 1"
confidence0
translations
Mathematica
translation"c == a + 1"
Maple
translation"c := a + 1"
positions
section3
sentence0
word20
includes
"a"
"c"
isPartOf
Empty array
definiens
definition"value"
score2
id41
pid91
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} \}"
confidence0.8614665289982916
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
Empty array
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} Will be translated to: Gamma[$0] Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E1 Mathematica: 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
Empty array
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} Will be translated to: GAMMA($0) Relevant links to definitions: DLMF: http://dlmf.nist.gov/5.2#E1 Maple: https://www.maplesoft.com/support/help/maple/view.aspx?path=GAMMA"
positions
section8
sentence0
word55
includes
"\,\Gamma(x)"
"\, \gamma"
"z"
"\,\gamma"
isPartOf
Empty array
definiens
definition"Euler"
score1
definition"Mascheroni"
score1
definition"gamma function"
score2
id42
pid92
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"
confidence0
translations
Mathematica
translation"192/24 == 8 == 2 * 4"
translationInformation
subEquations
"192/24 = 8"
"8 = 2 * 4"
freeVariables
Empty array
constraints
Empty array
tokenTranslations
\times"was translated to: *"
numericResults
overallResult"SUCCESS"
numberOfTests2
numberOfFailedTests0
numberOfSuccessfulTests2
numberOfSkippedTests0
numberOfErrorTests0
wasAbortedfalse
crashedfalse
testCalculationsGroups
lhs"192/24"
rhs"8"
testExpression"(192/24)-(8)"
activeConstraints
Empty array
testCalculations
result"SUCCESS"
resultExpression"0."
testValues
Empty object
lhs"8"
rhs"2 * 4"
testExpression"(8)-(2 * 4)"
activeConstraints
Empty array
testCalculations
result"SUCCESS"
resultExpression"0."
testValues
Empty object
symbolicResults
overallResult"SUCCESS"
numberOfTests2
numberOfFailedTests0
numberOfSuccessfulTests2
numberOfSkippedTests0
numberOfErrorTests0
crashedfalse
testCalculationsGroup
lhs"192/24"
rhs"8"
testExpression"(192/24)-(8)"
testCalculations
result"SUCCESS"
testTitle"Simple"
testExpression"FullSimplify[(192/24)-(8)]"
resultExpression"0"
wasAbortedfalse
conditionallySuccessfulfalse
lhs"8"
rhs"2 * 4"
testExpression"(8)-(2 * 4)"
testCalculations
result"SUCCESS"
testTitle"Simple"
testExpression"FullSimplify[(8)-(2 * 4)]"
resultExpression"0"
wasAbortedfalse
conditionallySuccessfulfalse
SymPy
translation"192/24 == 8 == 2 * 4"
translationInformation
subEquations
"192/24 = 8"
"8 = 2 * 4"
freeVariables
Empty array
constraints
Empty array
tokenTranslations
\times"was translated to: *"
Maple
translation"192/24 = 8 = 2 * 4"
translationInformation
subEquations
"192/24 = 8"
"8 = 2 * 4"
freeVariables
Empty array
constraints
Empty array
tokenTranslations
\times"was translated to: *"
numericResults
overallResult"SUCCESS"
numberOfTests2
numberOfFailedTests0
numberOfSuccessfulTests2
numberOfSkippedTests0
numberOfErrorTests0
wasAbortedfalse
crashedfalse
testCalculationsGroups
lhs"192/24"
rhs"8"
testExpression"evalf((192/24)-(8))"
activeConstraints
Empty array
testCalculations
result"SUCCESS"
resultExpression"0."
testValues
Empty object
lhs"8"
rhs"2 * 4"
testExpression"evalf((8)-(2 * 4))"
activeConstraints
Empty array
testCalculations
result"SUCCESS"
resultExpression"0."
testValues
Empty object
symbolicResults
overallResult"SUCCESS"
numberOfTests2
numberOfFailedTests0
numberOfSuccessfulTests2
numberOfSkippedTests0
numberOfErrorTests0
crashedfalse
testCalculationsGroup
lhs"192/24"
rhs"8"
testExpression"(192/24)-(8)"
testCalculations
result"SUCCESS"
testTitle"Simple"
testExpression"simplify((192/24)-(8))"
resultExpression"0"
wasAbortedfalse
conditionallySuccessfulfalse
lhs"8"
rhs"2 * 4"
testExpression"(8)-(2 * 4)"
testCalculations
result"SUCCESS"
testTitle"Simple"
testExpression"simplify((8)-(2 * 4))"
resultExpression"0"
wasAbortedfalse
conditionallySuccessfulfalse
positions
section3
sentence1
word25
includes
Empty array
isPartOf
Empty array
definiens
Empty array
id43
pid93
eid"math.93.0-1"
title"Gegenbauer polynomials"
formulae
id"FORMULA_34d9d355f0c0e28d91465c3b575fb0a1"
formula"=2"
semanticFormula"\alpha = 2"
confidence0
translations
Mathematica
translation"\[Alpha] = 2"
Maple
translation"alpha = 2"
positions
section1
sentence0
word17
includes
Empty array
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"
score2
id44
pid94
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}"
confidence0
translations
Empty object
positions
section1
sentence5
word13
includes
"q^{n}"
"q"
"b"
"a"
"z"
isPartOf
Empty array
definiens
definition"q-analog of the hypergeometric series"
score2
definition"unilateral basic hypergeometric series"
score2
definition"basic hypergeometric series"
score2
id45
pid95
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"
confidence0
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
section0
sentence2
word4
includes
"\mu"
"\kappa"
"z"
isPartOf
Empty array
definiens
definition"Whittaker 's equation"
score2
definition"Whittaker function"
score1
id46
pid96
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"
confidence0
translations
Mathematica
translation"Subscript[e, 1] == Divide[1,2] Subscript[e, 2] = 0 Subscript[e, 3] = -Divide[1,2]"
Maple
translation"e[1] := (1)/(2); e[2] := 0; e[3] := -(1)/(2)"
positions
section0
sentence5
word11
includes
"e_{1}"
"e_{2}"
"e_{3}"
isPartOf
Empty array
definiens
definition"constant"
score2
id47
pid98
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"
confidence0
translations
Mathematica
translation"\[Gamma] > 0 n - p == m - q > 0"
Maple
translation"gamma > 0; n - p = m - q > 0"
positions
section12
sentence0
word17
includes
"m"
"q"
"p=q> 0"
"n"
"p=q"
"\gamma>"
isPartOf
Empty array
definiens
definition"constraint"
score2
id48
pid99
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'}"
confidence0
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
section10
sentence0
word23
includes
"j"
"m"
isPartOf
Empty array
definiens
definition"Wigner 1-jm symbol"
score2
definition"metric tensor in angular-momentum theory"
score2
definition"quantity"
score0
id49
pid100
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}"
confidence0.8624533614429312
translations
Empty object
positions
section5
sentence4
word23
includes
"j"
isPartOf
Empty array
definiens
definition"6j symbol"
score2
definition"associativity isomorphism"
score2
definition"symbol"
score1
definition"vector space isomorphism"
score2
definition"Wigner"
score1
definition"Wigner 's 6 - j symbol"
score2
id50
pid101
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)}"
confidence0
translations
Empty object
positions
section5
sentence0
word10
includes
"j"
"_{4}"
isPartOf
Empty array
definiens
definition"orthogonality relation"
score1
definition"triangular delta"
score2
definition"symbol"
score1
definition"Wigner 's 9 - j symbol"
score2
id51
pid102
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}"
confidence0
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
section2
sentence0
word9
includes
"q"
"n"
isPartOf
Empty array
definiens
definition"following alternative expression"
score0
definition"Kravchuk polynomial"
score2
id52
pid103
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"
confidence0
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
section1
sentence1
word28
includes
"x"
"x)"
"g_1(x)"
isPartOf
Empty array
definiens
definition"series expansion"
score1
definition"special case"
score0
definition"asymptotic series"
score1
definition"definition"
score2
id53
pid104
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})"
confidence0.8775479393290169
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
section0
sentence1
word18
includes
"s_{\mu,\nu}(z)"
"S_{\mu,\nu}(z)"
"J_{\nu}(z)"
"Y_{\nu}(z)"
isPartOf
Empty array
definiens
definition"Lommel function"
score2
definition"Bessel function of the first kind"
score2
definition"Bessel function of the second kind"
score2
id54
pid105
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}}"
confidence0.8740850655136605
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
section6
sentence2
word31
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
Empty array
definiens
definition"hypergeometric function"
score2
definition"Struve"
score2
definition"Struve function"
score2
definition"gamma function"
score2
id55
pid106
eid"math.106.7"
title"Hill differential equation"
formulae
id"FORMULA_3a6745862e8f6ef2b93c343ad82b40c0"
formula"f(t+p) = f(t)"
semanticFormula"f(t+p) = f(t)"
confidence0
translations
Mathematica
translation"f[t + p] == f[t]"
Maple
translation"f(t + p) = f(t)"
positions
section0
sentence1
word21
includes
"f(t)"
"t"
"p"
"f(t+\pi)=f(t)"
isPartOf
"f(t+\pi)=f(t)"
definiens
definition"function"
score2
definition"periodic function by minimal period"
score2
id56
pid108
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}}}"
confidence0.8648813564530858
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
section2
sentence0
word8
includes
"J_{\nu}"
"\mathbf{J}_{\nu}"
"\nu"
isPartOf
Empty array
definiens
definition"power series expansion"
score2
definition"Anger function"
score2
definition"Gamma function"
score2
id57
pid109
eid"math.109.27"
title"Lamé 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"
confidence0
translations
Empty object
positions
section2
sentence3
word31
includes
"\operatorname{Ec}"
"\operatorname{Es}"
isPartOf
Empty array
definiens
definition"boundary condition"
score2
definition"ellipsoidal wave"
score2
id58
pid110
eid"math.110.1"
title"Gauss–Hermite 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)"
confidence0
translations
Empty object
positions
section0
sentence1
word3
includes
"\int_{-\infty}^{+\infty} e^{-x^2} f(x)\,dx"
"n"
"x_{i}"
"w_{i}"
isPartOf
Empty array
definiens
definition"value of integral"
score2
definition"form of Gaussian quadrature"
score2
definition"Gauss -- Hermite quadrature"
score2
definition"Hermite polynomial"
score1
definition"associated weight"
score2
id59
pid111
eid"math.111.0"
title"Askey–Wilson 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}"
confidence0
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
section0
sentence3
word4
includes
"\phi"
"_{n}"
"n"
isPartOf
Empty array
definiens
definition"basic hypergeometric function"
score2
definition"q-Pochhammer symbol"
score2
definition"Askey–Wilson polynomials"
score2
id60
pid112
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}"
confidence0.8953028732079359
translations
Mathematica
translation"Q[n_, x_, \[Alpha]_, \[Beta]_, N_] := HypergeometricPFQ[{- n , - x , n + \[Alpha]+ \[Beta]+ 1}, {\[Alpha]+ 1 , - N + 1}, 1]"
positions
section0
sentence3
word11
includes
"R_{n}(x;\gamma,\delta,N)"
"S_{n}(x;a,b,c)"
isPartOf
Empty array
definiens
definition"Hahn polynomial"
score2
definition"basic hypergeometric function"
score2
definition"hypergeometric function"
score2
id61
pid113
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"
confidence0
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
section0
sentence2
word5
includes
Empty array
isPartOf
Empty array
definiens
definition"orthogonality relation"
score2
definition"Charlier polynomial"
score2
id62
pid114
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}"
confidence0
translations
Empty object
positions
section1
sentence0
word15
includes
Empty array
isPartOf
Empty array
definiens
definition"term of basic hypergeometric function"
score2
id63
pid115
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}"
confidence0.5776294951318733
translations
Empty object
positions
section1
sentence0
word15
includes
Empty array
isPartOf
Empty array
definiens
definition"q-Charlier polynomial"
score2
definition"term of the basic hypergeometric function"
score2
id64
pid116
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}"
confidence0.8953028732079359
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
section0
sentence1
word16
includes
Empty array
isPartOf
Empty array
definiens
definition"Meixner polynomial"
score2
definition"Pochhammer symbol"
score1
definition"term of binomial coefficient"
score1
id65
pid117
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"
confidence0
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
section3
sentence0
word39
includes
"y"
"x"
"F_{1}"
"F"
"_{1}F_{1}"
isPartOf
Empty array
definiens
definition"Appell"
score2
definition"partial differential equation"
score2
definition"system of differential equation"
score1
definition"system of second-order differential equation"
score2
definition"Appell series"
score2
id66
pid118
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"
confidence0
translations
Mathematica
translation"\[CapitalTheta][\[CapitalLambda]_, \[Tau]_] := Sum[Exp[I*Pi*\[Tau]*(Norm[x])^(2)], {x, \[CapitalLambda]}]"
positions
section1
sentence0
word15
includes
"\Lambda"
isPartOf
Empty array
definiens
definition"theta function"
score2
definition"lattice"
score1
definition"Theta function of a lattice"
score1
id67
pid119
eid"math.119.0"
title"Heine–Stieltjes 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"
confidence0
translations
Empty object
positions
section0
sentence1
word6
includes
"V(z)"
"S"
"V"
isPartOf
Empty array
definiens
definition"form"
score0
definition"Fuchsian equation"
score2
definition"polynomial"
score1
definition"degree"
score0
definition"Edward Burr Van Vleck"
score0
definition"Heine"
score1
definition"polynomial solution"
score1
definition"Stieltjes polynomial"
score1
definition"Van Vleck polynomial"
score1
id68
pid120
eid"math.120.0"
title"Stieltjes–Wigert 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)"
confidence0
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
section0
sentence0
word38
includes
"\frac{k}{\sqrt{\pi}} x^{-1/2} \exp \left(-k^2 \log^2 x \right)"
isPartOf
Empty array
definiens
definition"weight function"
score2
definition"positive real line"
score0
definition"basic Askey scheme"
score1
definition"family of basic hypergeometric orthogonal polynomial"
score1
definition"mathematics"
score0
definition"Stieltjes -- Wigert polynomial"
score2
definition"Thomas Jan Stieltjes"
score0
definition"Carl Severin Wigert"
score0
definition"example of such weight function"
score0
id69
pid121
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)"
confidence0
translations
Mathematica
translation"(y)^(2) == x*(x - 1)*(x - \[Lambda])"
Maple
translation"(y)^(2) = x*(x - 1)*(x - lambda)"
positions
section2
sentence5
word13
includes
"\lambda"
isPartOf
Empty array
definiens
definition"elliptic curve of Legendre form"
score2
definition"relation to the j-invariant"
score1
definition"relation to the j-invariant"
score1
id70
pid122
eid"math.122.3"
title"Meixner–Pollaczek 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)"
confidence0.8953028732079359
translations
Empty object
positions
section1
sentence0
word9
includes
"P_{m}^{(\lambda)}(x;\varphi)"
isPartOf
Empty array
definiens
definition"first few Meixner -- Pollaczek polynomial"
score2
id71
pid123
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)}"
confidence0.7595006538205181
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
section1
sentence0
word12
includes
"P_{n}^{(\alpha, \beta)}(x)"
"(\alpha+1)_n"
"n"
"n + \alpha + \beta"
"P_{n}^{(\alpha, \beta)}"
"\alpha,\beta"
"z"
isPartOf
Empty array
definiens
definition"Pochhammer 's symbol"
score2
definition"hypergeometric function"
score2
definition"Jacobi polynomial"
score2
id72
pid124
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}"
confidence0.7132263353695951
translations
Empty object
positions
section0
sentence1
word10
includes
"R_{n}(x;\gamma,\delta,N)"
isPartOf
Empty array
definiens
definition"hypergeometric function"
score1
definition"dual Hahn polynomial"
score1
definition"continuous Hahn polynomial"
score1
definition"continuous dual Hahn polynomial"
score2
id73
pid125
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)}"
confidence0.9041995034970904
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
section5
sentence2
word20
includes
"p_{n}(x;a,b,c,d)"
"F_{n}"
"P_{n}^{(\alpha,\beta)}"
isPartOf
Empty array
definiens
definition"case of the continuous Hahn polynomial"
score1
definition"Jacobi polynomial"
score2
definition"continuous Hahn polynomial"
score2
id74
pid126
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"
confidence0
translations
Empty object
positions
section1
sentence0
word8
includes
"n"
isPartOf
Empty array
definiens
definition"Dual Hahn polynomial"
score2
id75
pid127
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}"
confidence0.8662724998444776
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
section1
sentence0
word15
includes
"q"
isPartOf
Empty array
definiens
definition"polynomial"
score1
definition"term of basic hypergeometric function"
score1
definition"Pochhammer symbol"
score1
definition"continuous FORMULA_7694f4a66316e53c8cdd9d9954bd611d - Hahn polynomial"
score2
definition"q - Pochhammer symbol"
score2
id76
pid128
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 , ae^{\iunit \theta} , ae^{- \iunit \theta}}{ab , ac}{q}{q}"
confidence0.8662724998444776
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*(e)^(I*\[Theta]), a*(e)^(- I*\[Theta])},{a*b , a*c},q,q]"
positions
section1
sentence0
word15
includes
"q"
isPartOf
Empty array
definiens
definition"polynomial"
score1
definition"term of basic hypergeometric function"
score2
definition"Pochhammer symbol"
score1
definition"continuous dual FORMULA_7694f4a66316e53c8cdd9d9954bd611d - Hahn polynomial"
score2
id77
pid129
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}"
confidence0
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
section1
sentence0
word15
includes
Empty array
isPartOf
Empty array
definiens
definition"q - Hahn polynomial"
score2
definition"polynomial"
score1
definition"term of basic hypergeometric function"
score2
definition"Pochhammer symbol"
score0
id78
pid131
eid"math.131.0"
title"Al-Salam–Chihara polynomials"
formulae
id"FORMULA_52a07ce46212cbc2298415c5fca6e075"
formula"x="
semanticFormula"x=\cos@{\theta}"
confidence0
translations
Mathematica
translation"x = Cos[\[Theta]]"
Maple
translation"x = cos(theta)"
positions
section1
sentence0
word20
includes
Empty array
isPartOf
Empty array
definiens
definition"cosine function"
score2
definition"substitution"
score2
id79
pid132
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})}}"
confidence0.7579553437219001
translations
Mathematica
translation"\[CapitalPhi]\[Prima][n_, z_] := z^n*Conjugate[\[CapitalPhi][n, Divide[1, Conjugate[z]]]]"
positions
section2
sentence0
word8
includes
"\Phi_n(z)"
"z^n"
"\alpha_n"
isPartOf
Empty array
definiens
definition"polynomial"
score2
id80
pid133
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}"
confidence0
translations
Empty object
positions
section5
sentence0
word16
includes
"P_{n}"
"c_{n}"
"P_{m}"
isPartOf
Empty array
definiens
definition"constant"
score0
definition"normalisation"
score0
definition"orthogonal polynomial"
score2
definition"term of the moment"
score0
id81
pid134
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}"
confidence0.7229065246531701
translations
Mathematica
translation"p[n_, x_, a_, b_, q_] := QHypergeometricPFQ[{(q)^(- n), a*b*(q)^(n + 1)},{a*q},q,x*q]"
positions
section1
sentence0
word19
includes
"q"
"p_{n}(x;a,b;q)"
isPartOf
Empty array
definiens
definition"Jacobi polynomial"
score1
definition"term of basic hypergeometric function"
score2
definition"Pochhammer symbol"
score0
definition"little q - Jacobi polynomial"
score2
id82
pid135
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}"
confidence0.7424814142326033
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
section1
sentence0
word11
includes
"P_{n}(x;a,b,c;q)"
"q"
isPartOf
Empty array
definiens
definition"polynomial"
score1
definition"term of basic hypergeometric function"
score2
definition"big q - Jacobi polynomial"
score2
id83
pid137
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}}"
confidence0
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
section1
sentence0
word15
includes
"q"
isPartOf
Empty array
definiens
definition"polynomial"
score1
definition"term of basic hypergeometric function"
score1
definition"Pochhammer symbol"
score1
definition"q - Pochhammer symbol"
score1
definition"big q - Laguerre polynomial"
score2
id84
pid138
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}"
confidence0
translations
Empty object
positions
section1
sentence0
word15
includes
"q"
isPartOf
Empty array
definiens
definition"polynomial"
score1
definition"term of basic hypergeometric function"
score2
definition"Pochhammer symbol"
score0
definition"dual q - Krawtchouk polynomial"
score2
id85
pid139
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}"
confidence0
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
section1
sentence1
word0
includes
"q"
isPartOf
Empty array
definiens
definition"continuous q - Laguerre polynomial"
score2
definition"family of basic hypergeometric orthogonal polynomial"
score2
definition"Pochhammer symbol"
score2
id86
pid142
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}"
confidence0.7219509974881755
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
section1
sentence0
word15
includes
"q"
"p_{n}(x;a|q)"
isPartOf
Empty array
definiens
definition"polynomial"
score1
definition"term of basic hypergeometric function"
score2
definition"Pochhammer symbol"
score1
definition"little q - Laguerre polynomial"
score2
definition"q - Pochhammer symbol"
score2
id87
pid143
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}"
confidence0.6264217257193126
translations
Mathematica"y[n_, x_, a_, q_] := QHypergeometricPFQ[{(q)^(- N), - a*(q)^(n)},{0},q,q*x]"
positions
section1
sentence0
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includes
Empty array
isPartOf
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definiens
definition"polynomial"
score1
definition"term of basic hypergeometric function"
score2
definition"Pochhammer symbol"
score0
definition"q - Bessel polynomial"
score1
id88
pid144
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}"
confidence0.8429359579302446
translations
Empty object
positions
section1
sentence0
word27
includes
"\hat{h}_{n}(x;q)"
"q"
"h_{n}(x;q)"
isPartOf
Empty array
definiens
definition"Hermite polynomial"
score2
definition"term of basic hypergeometric function"
score1
definition"Carlitz polynomial"
score1
definition"Al-Salam"
score1
definition"discrete q - Hermite polynomial"
score2
id89
pid145
eid"math.145.0"
title"Q-Meixner–Pollaczek 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}"
confidence0.8662724998444776
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
section1
sentence0
word16
includes
Empty array
isPartOf
Empty array
definiens
definition"polynomial"
score1
definition"term of basic hypergeometric function"
score2
definition"Pochhammer symbol"
score1
definition"q - Pochhammer symbol"
score2
definition"Q Meixner – Pollaczek polynomials"
score2
id90
pid149
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}"
confidence0.779734956061429
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
section1
sentence0
word18
includes
"q"
"P_{n}^{(\alpha)}(x;q)"
isPartOf
Empty array
definiens
definition"Laguerre polynomial"
score2
definition"q - Laguerre polynomial"
score2
definition"term of basic hypergeometric function"
score2
definition"Pochhammer symbol"
score1
definition"q - Pochhammer symbol"
score2
id91
pid150
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}}"
confidence0.7796357038819148
translations
Empty object
positions
section3
sentence0
word0
includes
"q"
isPartOf
Empty array
definiens
definition"continuous q - Hermite polynomial"
score2
definition"q - Pochhammer symbol"
score2
id92
pid151
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"
confidence0
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
section0
sentence0
word19
includes
"p"
isPartOf
Empty array
definiens
definition"differential equation"
score2
definition"Ince equation"
score2
definition"mathematics"
score0
definition"non-negative integer"
score0
definition"Edward Lindsay Ince"
score0
definition"polynomial solution"
score0
definition"Ince polynomial"
score1
id93
pid152
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}}"
confidence0.8133162393162393
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
section1
sentence0
word13
includes
Empty array
isPartOf
Empty array
definiens
definition"Ferrers function of the second kind"
score2
definition"Ferrers function of the first kind"
score1
definition"Gamma function"
score2
definition"hypergeometric function"
score2
id94
pid153
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)"
confidence0
translations
Empty object
positions
section2
sentence0
word16
includes
"v"
"w"
"H_v^{(1)}(z,w)"
isPartOf
Empty array
definiens
definition"incomplete Bessel function"
score2
id95
pid154
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}"
confidence0
translations
Mathematica
translation"K[v_, x_, y_] := Integrate[Divide[Exp[- x*t -Divide[y,t]],(t)^(v + 1)], {t, 1, Infinity}, GenerateConditions->None]"
positions
section0
sentence0
word18
includes
"K_v(x,y)"
isPartOf
Empty array
definiens
definition"mathematician"
score0
definition"type incomplete-version of Bessel function"
score2
definition"type generalized-version of incomplete gamma function"
score0
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