Generalized Laguerre polynomials: Integration (formula 05.08.21.0010)

LaguerreL

$Mathematica Notation$

Polynomials

LaguerreL[n,lambda,z]

Integration

Definite integration

Involving the direct function

http://functions.wolfram.com/05.08.21.0010.01

Input Form

Integrate[(t^(\[Alpha] - 1) LaguerreL[m, \[Lambda], p t] LaguerreL[n, \[Beta], p t])/E^(p t), {t, 0, Infinity}] == ((Gamma[\[Alpha]] Gamma[n - \[Alpha] + \[Beta] + 1] Gamma[m + \[Lambda] + 1])/(p^\[Alpha] (m! n! Gamma[-\[Alpha] + \[Beta] + 1] Gamma[\[Lambda] + 1]))) HypergeometricPFQ[{-m, \[Alpha], \[Alpha] - \[Beta]}, {-n + \[Alpha] - \[Beta], \[Lambda] + 1}, 1] /; Re[\[Alpha]] > 0 && Re[p] > 0 && Element[m, Integers] && m >= 0 && Element[n, Integers] && n >= 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)

2007-05-02

LaguerreL[n,z]
LaguerreL[nu,z]
LaguerreL[nu,mu,z]