I'm struggling with this question, which says "given the generating funcion $g(x,z)=e^{-z^2 + 2xz} = \sum_{n=0}^{\infty}H_n(x) \frac{z^n}{n!}$ prove that the Hermite polynomials satisfy the Hermite equation".
So far I tried using the Rodrigues' expresion (which I already derived from the generating function, and it's the most explicit formula I could find for $H_n$) and tried to plug it into the Hermite equation, but I wasn't succesful. Is this the way to work it out? Any ideas?
Thanks a lot!