Reference request for a lemma on the expected value of Hermitian polynomials of Gaussian random variables.

Question

I am looking for a reference of the following lemma, specifically a textbook on which I can self-study the statement and the proof

If $(U,V)$ a gaussian vector such that $U,V \sim N(0,1)$ then for all $p,q \in N$

$$E[H_q(U) H_p(V)]= q! E[UV]^q $$

If $p=q$. ( $0$ if $p \ne q$). Where $H_n$ is the n-th Hermitian polynomial.

Answer 1

This is treated for instance in Chapter 11.2 of Analysis of Boolean Functions, by Ryan O'Donnell. (Not the most standard reference for this fact, I assume, but a convenient one as the book can be consulted online).

You can find a version online here: http://www.contrib.andrew.cmu.edu/~ryanod/?p=1694 (look at the derivation leading to (5), and providing the proof of Proposition 31).