# generating function of Hermite polynomials

We start from the definition of Hermite polynomials via their http://planetmath.org/node/11983Rodrigues formula

 $\displaystyle H_{n}(z)\;:=\;(-1)^{n}e^{z^{2}}\frac{d^{\,n}}{dz^{n}}e^{-z^{2}}% \qquad(n\;=\;0,\,1,\,2,\,\ldots).$ (1)

The consequence

 $\displaystyle f^{(n)}(z)\;=\;\frac{n!}{2\pi i}\oint_{C}\frac{f(\zeta)}{(\zeta-% z)^{n+1}}\ d\zeta$ (2)

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

 $H_{n}(z)\;=\;(-1)^{n}\frac{n!}{2i\pi}\oint_{C}\frac{e^{z^{2}-\zeta^{2}}}{(% \zeta\!-\!z)^{n+1}}\,d\zeta,$

where $C$ is any contour around the point $z$ and the direction is anticlockwise.  The http://planetmath.org/node/11373substitution  $z\!-\!\zeta:=t$  here yields

 $H_{n}(z)\;=\;\frac{n!}{2i\pi}\oint_{C^{\prime}}\frac{e^{z^{2}-(z-t)^{2}}}{t^{n% +1}}\,dt,$

where the contour $C^{\prime}$ goes round the origin.  Accordingly, by (2) we can infer that

 $H_{n}(z)\;=\;\left[\frac{d^{\,n}}{dt^{n}}e^{z^{2}-(z-t)^{2}}\right]_{t=0},$

whence we have found the generating function

 $e^{z^{2}-(z-t)^{2}}\;=\;\sum_{n=0}^{\infty}H_{n}(z)\frac{t^{n}}{n!}$

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