Suppose I have expanded the function $f(x)$ in Hermite polynomials, \begin{equation} a_n = \int_{-\infty}^\infty f(x)\exp(-x^2)\mathcal{H}_n(x)dx \,. \end{equation}
How can I express the expansion coefficients for $f(\sqrt{x^2+\alpha^2})$ in terms of coefficients $a_n$?