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 $\{H\_S(x)\}\_{S}$?



Hermite polynomial after rotation
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](https://jscholarship.library.jhu.edu/bitstream/handle/1774.2/60082/REYNOLDS-DISSERTATION-2018.pdf?sequence=1&isAllowed=y). See also [Accurate Image Rotation using Hermite Expansions.](http://europepmc.org/backend/ptpmcrender.fcgi?accid=PMC2837196&blobtype=pdf)


