Let $H_n$ be the $n$th Hermite polynomial of the probabilist. For example, $H_0(x) = 1$, $H_1(x)=x$, $H_2(x) = x^2 - 1$, $H_3(x) = x^3-3x$, etc. Let $u$ and $v$ be unit-vectors in $\mathbb R^d$, and let $X=(x_1,\ldots,x_n)$ be uniform on the unit-sphere in $\mathbb R^d$.
Question. Given any pair of nonnegative integers $m,n$, what is the value of $c_{n,m}(u,v)=\mathbb E_{X}[H_n(X^\top u)H_m(X^\top v)]$ as a function of $m$, $n$, and $u^\top v$?
Notes
- Thanks to rotational invariances $c_{n,m}(u,v)$ equals $\mathbb E[H_n(x_1)H_n(tx_1 + (1-t^2)^{1/2}x_2)]$, with $t=u^\top v$.
- Because $X$ and $-X$ are equal in distribution, $i$ and $j$ have different parities, then $\mathbb E_X[(X^\top u)^i (X^\top v)^j] = 0$.
- I'm fine with estimates of $c_{n,m}(u,v)$ which are correct up to within an error of $\mathcal O(1/d^3)$ for large $d$.
Update
Question has been fully answered here https://mathoverflow.net/a/405901/78539.