Probabilists' Hermite polynomial (${\mathrm{He}}_n$)

The probabilists' Hermite polynomials areThey are orthogonal for the standard Gaussian weight and satisfy ${\mathrm{He}}_{n+1}=x{\mathrm{He}}_n-n{\mathrm{He}}_{n-1}$.

This is the Gaussian-probability normalization of the Hermite polynomial family.

The space-time Probabilists' Hermite polynomial $H_n(x,t)=t^{n/2}{\mathrm{He}}_n(x/\sqrt{t})$ extends polynomially to $t=0$ and solves ${\partial }_t{H}_n+\frac{1}{2}{\partial }_x^2{H}_n=0$. Therefore $H_n(B_t,t)$ is a martingale for a standard Brownian motion, by the Itô formula and Gaussian moments.