Hermite polynomial after rotation

Question

When we consider the $n$-dimensional standard normal distribution, the orthogonal basis is $\{H_S(x)\}_{S}$ where $H_S(x) = \prod_{k=1}^n H_{s_k}(x_k)$. Here $H_*(x)$ is the normalized probabilist's hermite polynomial. Suppose $U$ is any real orthogonal matrix. How to express $H_S(Ux)$ in terms of $\{H_S(x)\}_{S}$?

Answer 1

The expansion coefficients of a function $f(x_1,x_2,\ldots x_n)$ in the rotated basis of Hermite polynomials are related to the original expansion coefficients by an orthogonal matrix of "steering coefficients". Explicit expressions for $n=1,2,3$ are given in section 3.6 of K.L. Reynolds, Convolution, Rotation, and Data Fusion with Orthogonal Expansions. See also Accurate Image Rotation using Hermite Expansions.