Probabilists' Hermite polynomial
= Probabilists' Hermite polynomial
{c}
{title2=$\mathrm{He}_n$}
The probabilists' Hermite polynomials are
$$
\mathrm{He}_n(x)=(-1)^ne^{x^2/2}\frac{d^n}{dx^n}e^{-x^2/2}.
$$
They 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.