generating function of Laguerre polynomials

We start from the definition of Laguerre polynomialsDlmfDlmfDlmfMathworldPlanetmath via their formulaPlanetmathPlanetmath

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

The consequence

f(n)(z)=n!2πiCf(ζ)(ζ-z)n+1𝑑ζ (2)

of integral formulaPlanetmathPlanetmath allows to write (1) as the complex integral


where C is any contour around the point z and the direction is anticlockwise.  The

ζ-z:=zt1-t,ζ=z1-t,t= 1-zζdζ=zdt(1-t)2

here yields


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


whence we have found the generating function


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