** Generating Function (5-30 min)

We define the generating function of the Hermite polynomials as

$$e^{2tx-t^2}=\sum_{n=0}^{\infty}\frac{H_n(x)}{n!}t^n,\quad -\infty<x,t<\infty .......$$ (36)

Prove the formula of Rodrigues,

$$H_n(x)=(-1)^ne^{x^2}\frac{d^n}{dx^n}(e^{-x^2}).$$

Hint: Differentiate with respect to $t$.