This question is related to: <https://math.stackexchange.com/q/4270522/168758>




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Let $H\_n(x) \in \mathbb R[x]$ be the probabilist's $n$th Hermite polynomial. This an $n$th degree polynomial given by the following equivalent formulae (which ever helps)


$$
\begin{split}
H\_n(x) &= n!\sum\_{k=0}^{\lfloor n/2\rfloor}\frac{(-1)^k}{2^kk!(n-2k)!}x^{n-2k}\\
H\_n(x) &= \frac{1}{\sqrt{2\pi}}\int\_{-\infty}^\infty (x+iy)^n e^{-y^2/2}dy\\
H\_n(x) &= e^{-D^2/2}x^n,
\end{split}
$$
where $D^2$ is the second-derivative-w.r.t-$x$ differential operator $\dfrac{d^2}{dx^2}$, and $e^{-D^2/2}$ should be seen as a power series in $D^2$.


Let $d$ be a large positive integer, $a$ and $b$ be fixed vectors on the unit $(d-1)$-dimensional sphere $S\_{d-1}$, and $X$ be uniformly distributed on $S\_{d-1}$. For fixed nonnegative integers $n$ and $m$, define


$$
s\_{n,m} = \mathbb E[H\_n(X^\top a)H\_m(X^\top b)].
$$
Due to rotational-invarfiance of $X$, it is clear that $s\_{n,m}$ is a polynomial in $t:=a^\top b$. Let $c\_{n,m,k}$ be the coefficient of $t^k$ in $s\_{n,m}$.



> 
> **Question.**
> For $k \ge 1$, what is a good Big-O upper-bound for $c\_{n,m,k}$ in the limit $d \to \infty$.
> 
> 
> 



Integral of product of Hermite polynomials w.r.t marginal distribution of first two-coordinate of random vector on unit-sphere
To find the dependence of $s\_{nm}$ on $t=a\cdot b$, we take $a=(t,\sqrt{1-t^2},0,0,\ldots 0)$, $b=(1,0,0,0,\ldots 0)$, so that
$$s\_{nm} = \mathbb E[H\_n(X^\top a)H\_m(X^\top b)]=\mathbb E[H\_n(X\_1 t+X\_2\sqrt{1-t^2})H\_m(X\_1)].$$
The marginal distribution $P(X\_1,X\_2)$ of two elements from a vector that is uniformly distributed on the $d$-dimensional unit sphere is given by (see, for example, this [calculation](https://stats.stackexchange.com/a/520811))
$$P(X\_1,X\_2)=\frac{d-2}{2\pi}(1-X\_1^2-X\_2^2)^{d/2-2},\;\;X\_1^2+X\_2^2<1,\;\;d\geq 3.$$
Hence we have for $s\_{nm}$ the integral expression
$$s\_{nm}=\frac{d-2}{2\pi}\int\_{0}^{1}rdr\int\_0^{2\pi}d\phi\, (1-r^2)^{d/2-2}H\_n\left(rt\cos\phi+r\sqrt{1-t^2}\sin\phi\right)H\_m(r\cos\phi).$$


For large $d$ the Hermite polynomials can be expanded around $r=0$, which gives
$$s\_{nm}\approx \frac{\pi}{d} 2^{\frac{1}{2} (m+n-2)} \left(\frac{4 t}{\Gamma \left(-\frac{m}{2}\right) \Gamma \left(-\frac{n}{2}\right)}-\frac{-2 d+m+n}{\Gamma \left(\frac{1}{2}-\frac{m}{2}\right) \Gamma \left(\frac{1}{2}-\frac{n}{2}\right)}\right),\;\;d\gg 1.$$


