generating function of Laguerre polynomials


We start from the definition of Laguerre polynomialsDlmfDlmfDlmfMathworldPlanetmath via their http://planetmath.org/node/11983Rodrigues formulaPlanetmathPlanetmath

Ln(z):=ezdndzne-zzn  (n= 0, 1, 2,). (1)

The consequence

f(n)(z)=n!2πiCf(ζ)(ζ-z)n+1𝑑ζ (2)

of http://planetmath.org/node/1150Cauchy integral formulaPlanetmathPlanetmath allows to write (1) as the complex integral

Ln(z)=n!2iπCeze-ζ(ζ-z)n+1𝑑ζ=n!2iπCez-ζdζ(1-zζ)n(ζ-z),

where C is any contour around the point z and the direction is anticlockwise.  The http://planetmath.org/node/11373substitution

ζ-z:=zt1-t,ζ=z1-t,t= 1-zζdζ=zdt(1-t)2

here yields

Ln(z)=n!2iπCe-zt1-tzdt(1-t)2tnzt1-t=n!2iπCe-zt1-tdt(1-t)tn+1

where the contour C goes round the origin.  Accordingly, by (2) we can infer that

Ln(z)=[dndtne-zt1-t1-t]t=0,

whence we have found the generating function

e-zt1-t1-t=n=0Ln(z)n!tn

of the Laguerre polynomials.

Title generating function of Laguerre polynomials
Canonical name GeneratingFunctionOfLaguerrePolynomials
Date of creation 2013-03-22 19:06:51
Last modified on 2013-03-22 19:06:51
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Derivation
Classification msc 33B99
Classification msc 30B10
Classification msc 26C05
Classification msc 26A09
Classification msc 33E30
Related topic ExampleOfFindingTheGeneratingFunction
Related topic GeneratingFunctionOfHermitePolynomials
Related topic VariantOfCauchyIntegralFormula