Gold 13

Hermite polynomials

Gold ID

13

Link

https://sigir21.wmflabs.org/wiki/Hermite_polynomials#math.63.109

Formula

$E(x,y;u):=\sum _{n=0}^{\infty }u^{n}\,\psi _{n}(x)\,\psi _{n}(y)={\frac {1}{\sqrt {\pi (1-u^{2})}}}\,\exp \left(-{\frac {1-u}{1+u}}\,{\frac {(x+y)^{2}}{4}}-{\frac {1+u}{1-u}}\,{\frac {(x-y)^{2}}{4}}\right)$

TeX Source

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)

Translation Results
Semantic LaTeX	Mathematica Translation	Maple Translations

Semantic LaTeX

Translation

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})

Expected (Gold Entry)

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})

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]]

Expected (Gold Entry)

\[CapitalEpsilon][x_, y_, u_] := Sum[(u)^(n)* Subscript[\[Psi], n][x]* Subscript[\[Psi], n][y], {n, 0, Infinity}] == 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))

Expected (Gold Entry)

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))