Editing Hermite polynomials (section)

===Generating function===
The Hermite polynomials are given by the [[exponential generating function]]
<math display="block">\begin{align}
 e^{xt - \frac12 t^2} &= \sum_{n=0}^\infty \operatorname{He}_n(x) \frac{t^n}{n!}, \\
 e^{2xt - t^2} &= \sum_{n=0}^\infty  H_n(x) \frac{t^n}{n!}.
\end{align}</math>

This equality is valid for all [[complex number|complex]] values of {{mvar|x}} and {{mvar|t}}, and can be obtained by writing the Taylor expansion at {{mvar|x}} of the entire function {{math|''z'' → ''e''<sup>−''z''<sup>2</sup></sup>}} (in the physicist's case). One can also derive the (physicist's) generating function by using [[Cauchy's integral formula]] to write the Hermite polynomials as
<math display="block">H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} = (-1)^n e^{x^2} \frac{n!}{2\pi i} \oint_\gamma \frac{e^{-z^2}}{(z-x)^{n+1}} \,dz.</math>

Using this in the sum 
<math display="block">\sum_{n=0}^\infty H_n(x) \frac {t^n}{n!},</math> 
one can evaluate the remaining integral using the calculus of residues and arrive at the desired generating function.

A slight generalization states<ref>(Rainville 1971), p. 198</ref><math display="block">e^{2 x t-t^2} H_k(x-t) = \sum_{n=0}^{\infty} \frac{H_{n+k}(x) t^n}{n!}</math>