Let $d$ be a large positive integer and $t \in [-1,1]$. For any pair of integers $n,m \ge 0$ of same parity, define $s_{nm}$ by $$ s_d(n,m)=\frac{d-2}{\pi}\int_{0}^{1}rdr\int_0^{2\pi}d\phi\, (1-r^2)^{d/2-2}H_n\left(rt\cos\phi+r\sqrt{1-t^2}\sin\phi\right)H_m(r\cos\phi). $$ where $H_n(z)$ is the probabilist's $n$th Hermite polynomial. Note that $s_{nm}$ is itself a polynomial in $t$ (of degree at most $\min(m,n)$). Finally let $k$ be a positive integer $\le \min(m,n)$.
Question. As a function of $n$, $m$, $k$, and $d$, what is a good estimate (valid for large $d$) for the coefficient $s_d(n,m)[t^k]$ of $t^k$ in $s_d(n,m)$ ?
Solution for $k = 0$ and $k=1$
Via an order-$1$ Taylor expansion of $H_n(rt\cos\phi + r \sqrt{1-t^2}\sin\phi)H_m(r\cos\phi)$ around $r=0$, it was shown in this MO post that for $k=0$, we have $$ s_d(n,m)[t^0] = \begin{cases} H_n(0)H_m(0)+(1/d)(H_n(0)H_{m+1}(0)+H_{n+1}(0)H_m(0)),&\mbox{ if }n,m\text{ even },\\ 0,&\mbox{ else.} \end{cases} $$ Likewise, for $k=1$, we have $$ s_d(n,m)[t^1] = \begin{cases}(1/d)H_{n+1}(0)H_{m+1}(0),&\mbox{ if }n,m\text{ odd },\\ 0,&\mbox{ else.} \end{cases} $$