Gold 31

Laguerre polynomials

Gold ID: 31
Link: https://sigir21.wmflabs.org/wiki/Laguerre_polynomials#math.81.84
Formula: $\sum _{n=0}^{\infty }{\frac {n!\,\Gamma \left(\alpha +1\right)}{\Gamma \left(n+\alpha +1\right)}}L_{n}^{(\alpha )}(x)L_{n}^{(\alpha )}(y)t^{n}={\frac {1}{(1-t)^{\alpha +1}}}e^{-(x+y)t/(1-t)}\,_{0}F_{1}\left(;\alpha +1;{\frac {xyt}{(1-t)^{2}}}\right)$
TeX Source: \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)

Translation Results
Semantic LaTeX	Mathematica Translation	Maple Translations

Translation: \sum_{n=0}^\infty \frac{n! \Gamma(\alpha + 1)}{\Gamma(n + \alpha + 1)} \LaguerrepolyL[\alpha]{n}@{x} \LaguerrepolyL[\alpha]{n}@{y} t^n = \frac{1}{(1-t)^{\alpha + 1}} e_0^{-(x+y)t/(1-t)} F_1(; \alpha + 1 ; \frac{xyt}{(1-t)^2})
Expected (Gold Entry): \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}}

Translation: Sum[Divide[(n)!*\[CapitalGamma]*(\[Alpha]+ 1),\[CapitalGamma]*(n + \[Alpha]+ 1)]*LaguerreL[n, \[Alpha], x]*LaguerreL[n, \[Alpha], y]*(t)^(n), {n, 0, Infinity}, GenerateConditions->None] == Divide[1,(1 - t)^(\[Alpha]+ 1)]*(Subscript[e, 0])^(-(x + y)*t/(1 - t))*Subscript[F, 1][; \[Alpha]+ 1 ;Divide[x*y*t,(1 - t)^(2)]]
Expected (Gold Entry): Sum[Divide[(n)!*Gamma[\[Alpha]+ 1],Gamma[n + \[Alpha]+ 1]]*LaguerreL[n, \[Alpha], x]*LaguerreL[n, \[Alpha], x]*(t)^(n), {n, 0, Infinity}] == Divide[1,(1 - t)^(\[Alpha]+ 1)]*Exp[-(x + y)*t/(1 - t)]*HypergeometricPFQ[{}, {\[Alpha]+ 1}, Divide[x*y*t,(1 - t)^(2)]]

Translation: sum((factorial(n)*Gamma*(alpha + 1))/(Gamma*(n + alpha + 1))*LaguerreL(n, alpha, x)*LaguerreL(n, alpha, y)*(t)^(n), n = 0..infinity) = (1)/((1 - t)^(alpha + 1))*(e[0])^(-(x + y)*t/(1 - t))*F[1](; alpha + 1 ;(x*y*t)/((1 - t)^(2)))
Expected (Gold Entry): 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)))