The Hermite functions $\phi_n$ obey the recursion relation
$$x \phi_n = \sqrt{\frac{n}{2}} \phi_{n-1} + \sqrt{\frac{n+1}{2}} \phi_{n+1}.$$
Through repeated use of this relation, we find the sum
$$x^m \phi_{n_0} = \sum_{n=0}^\infty c_n \phi_n$$
where most $c_n$ are zero. How do I derive a general formula for $c_{n}$ as a function of $m$ and $n_0$? I know that the number of terms in each $c_n$ obey Pascal's triangle, such that e.g. for $c_{n_0}$ at $m=4$ there are $\binom{4}{2}=6$ terms, while for $c_{n_0\pm 1}$ there are $\binom{4}{1} = \binom{4}{3}=4$ terms. I also know that the outermost non-zero terms (i.e. $c_{n \pm m}$) involve falling and rising factorials, but I am at a loss for how to derive a closed expression for $c_n$. Can you help me on the right path?
I suspect that this is something that can be looked up somewhere, but so far I have not had any luck. Any help for search terms or direct sources would be much appreciated.