Generalized Laguerre polynomials: Summation (formula 05.08.23.0013)

LaguerreL

$Mathematica Notation$

Polynomials

LaguerreL[n,lambda,z]

Summation

Infinite summation

http://functions.wolfram.com/05.08.23.0013.01

Input Form

Sum[((t (1 - t)^\[Mu] E^((a z t)/(1 - t)))^k/(k + 1)) LaguerreL[k, \[Lambda] + \[Mu] k, z (1 + a k)], {k, 0, Infinity}] == (1/(t (\[Lambda] - \[Mu]))) ((E^(((-z) t + z - a z)/(1 - t)) (Hypergeometric1F1[\[Lambda] - \[Mu], \[Lambda] - \[Mu] + 1, (z (a - 1))/(1 - t)] - (1 - t)^(\[Lambda] - \[Mu]) Hypergeometric1F1[\[Lambda] - \[Mu], \[Lambda] - \[Mu] + 1, z (a - 1)]))/(1 - t)^\[Lambda]) /; Abs[t (1 - t)^\[Mu] E^((a z t)/(1 - t))] < 1

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

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