Gold 61

Charlier polynomials

Gold ID

61

Link

https://sigir21.wmflabs.org/wiki/Charlier_polynomials#math.113.2

Formula

$\sum _{x=0}^{\mathrm{\infty }}\frac{{\mu }^x}{x!}{C}_n(x;\mu )C_m(x;\mu )={\mu }^{-n}{e}^{\mu }n!{\delta }_{nm},\mu >0$

TeX Source

\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

Translation Results
Semantic LaTeX	Mathematica Translation	Maple Translations
		-

Semantic LaTeX

Translation

\sum_{x=0}^\infty \frac{\mu^x}{x!} C_n(x ; \mu) C_m(x ; \mu) = \mu^{-n} \expe^\mu n! \delta_{nm} , \quad \mu > 0

Expected (Gold Entry)

\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

Mathematica

Translation

Sum[Divide[\[Mu]^(x),(x)!]*Subscript[C, n][x ; \[Mu]]* Subscript[C, m][x ; \[Mu]], {x, 0, Infinity}, GenerateConditions->None] == \[Mu]^(- n)* Exp[\[Mu]]*(n)!*Subscript[\[Delta], n, m]

Expected (Gold Entry)

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]

Maple

Translation

sum(((mu)^(x))/(factorial(x))*C[n](x ; mu)* C[m](x ; mu), x = 0..infinity) = (mu)^(- n)* exp(mu)*factorial(n)*delta[n, m]

Expected (Gold Entry)