generating function of Hermite polynomials

We start from the definition of Hermite polynomialsDlmfDlmfDlmfMathworldPlanetmath via their formulaPlanetmathPlanetmath

Hn(z):=(-1)nez2dndzne-z2  (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  z-ζ:=t  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 Hermite polynomials.

Title generating function of Hermite polynomials
Canonical name GeneratingFunctionOfHermitePolynomials
Date of creation 2013-03-22 19:05:25
Last modified on 2013-03-22 19:05:25
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Derivation
Classification msc 33E30
Classification msc 33B99
Classification msc 26C05
Classification msc 26A09
Classification msc 12D99
Related topic OrthogonalPolynomials
Related topic ExampleOfFindingTheGeneratingFunction
Related topic GeneratingFunctionOfLaguerrePolynomials
Related topic VariantOfCauchyIntegralFormula