Let $n,m \ge 0$ be integers, $t \in [-1,1]$ and $r \in [0,1]$, and define $$ s_{n,m}(t) := \int_0^{2\pi}H_n(rt\cos\phi+r(1-t^2)^{1/2}\sin\phi)H_m(r\cos\phi) d\phi, $$ where $H_n$ is the $n$th Hermite polynomial. Note that $s_{n,m}(t)$ is a polynomial in $t$.
Question. For any integer $k \in \{0,1,2,3\}$, what is the coefficient $s_{n,m}[t^k]$ of $t^k$ in $s_{n,m}(t)$ ?
Observations
By basic properties of polynomials, we have the formula $$ s_{n,m}(t)[t^k] = D^k s_{n,m}(t)\mid_{t=0}, $$ where $D$ is the differential operator $d/dt$.
Thus, one computes $s_{n,m}(t)[t^k] = \int_0^{2\pi}F_{n,k}(\phi)H_m(r\cos\phi) d\phi$, where
- $F_{n,0}(\phi) = H_n(r\sin\phi)$
- $F_{n,1}(\phi) = r\cos\phi H_n(r\sin\phi)$
- $F_{n,2}(\phi) = 4n(n-1)r^2\cos^2\phi H_{n-2}(r\sin\phi)-2nr\sin\phi H_{n-1}(r\sin\phi)$
- ...
(with the convention that $H_k(z) \equiv 0$ if $k < 0$)
Thus, the issue is reduced to being able to compute integrals of the form
$$ I = \int_0^{2\pi} \sin^N\phi H_n(r\sin\phi)\cos^M\phi H_m(r\cos\phi)d\phi. $$