Are there well know properties of \begin{align} E[ H_n(V)] \end{align} where $H_n( \cdot)$ is a Hermite polynomial and $V \in L^n$?
For example if $V$ is Gaussian with variance $1$ and mean $\mu$ than \begin{align} E[ H_n(V)]=\mu^n \end{align}
Also a reference for beginners would be great.
Thank you