Hermite polynomials: Integration (formula 05.01.21.0023)

HermiteH

$Mathematica Notation$

Involving the direct function

http://functions.wolfram.com/05.01.21.0023.01

Input Form

Integrate[E^(-(z^2/2) - (1/2) (z - \[Zeta])^2) HermiteH[n, z] HermiteH[p, z - \[Zeta]], {z, -Infinity, Infinity}] == (((Sqrt[Pi] Sqrt[2^n n!] Sqrt[2^p p!] Sqrt[2^(n - p) Gamma[1 + n] Gamma[1 + p]])/Gamma[1 + n]) (-\[Zeta])^(-n + p) Hypergeometric1F1Regularized[-n, 1 - n + p, \[Zeta]^2/2])/ E^(\[Zeta]^2/4) /; Element[n, Integers] && n >= 0 && Element[p, Integers] && p >= 0

Standard Form

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MathML Form

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Rule Form

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Date Added to functions.wolfram.com (modification date)

2002-12-18

HermiteH[nu,z]