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

---

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 $d \to \infty$.