I am embedding a sequence $\{a_n\}$ in a generating function using Hermite polynomials:
\begin{align} \{a_n\} \mapsto f_{\{a_n\}}(x)=\sum_n H_n(x)a_n \end{align}
I would like to find an inverse transformation from $f_{\{a_n\}}(x)$ to a given $a_k$.
The only thing that I came up with so far is to use the orthogonality of Hermite polynomials, so that
\begin{align} \int_{-\infty}^\infty \frac{H_k(x)e^{-x^2}}{\sqrt{\pi}2^k k!}f_{\{a_n\}}(x)dx = a_k \end{align}
However, integrals introduce issues in other parts of my project and I would like to avoid them. Are there simpler alternatives (for instance a differential operator)?
Suppose we have a sequence $\,\{a_n\}_{n=0}^\infty.\,$ Define a function $\,f(x)\,$ by $$ f(x) := \sum_{n=0}^\infty H_n(x)\,a_n \tag{1} $$ assuming that $\,f(x)\,$ is convergent and analytic at $0$. The given question asks
We need to express $\,x^n\,$ as a linear combination of Hermite polynomials. The OEIS sequence A067147 solves that problem. Accordingly, define the function
$\, A(n,k) := n!/(k!\,((n-k)/2)!\,2^n) \,$ if $n\ge 0,\; k\le n,\,$ and $n-k$ is even. Otherwise, $\,A(n,k)=0.$
This function has the desired property that $$ x^n = \sum_{k=0}^n H(k)\,A(n,k). \tag{2} $$ Get the power series coefficients of $\,f(x)\,$ using $$ b_n :=(d^n f(x)/dx^n)/n!, \qquad f(x) = \sum_{n=0}^\infty b_n\,x^n. \tag{3}$$ Finally we get the $$ a_k = \sum_{n=0}^\infty A(n,k)\,b_n. \tag{4} $$ The exponential generating function of $\,A(n,k)\,$ gives the ordinary generating function of $\,\{a_n\}$ $$ g(y) := \sum_{k=0}^\infty a_k\,y^k = \exp(\,y\, \mathcal{D}+ \mathcal{D}^2)[f(x)] \tag{5} $$ where the differential operator $\,\mathcal{D}\,$ is $$ \mathcal{D}[f(x)]:=\frac12 \frac{df(x)}{dx}. \tag{6} $$