Infinite sum of normalized Hermite polynomials

Question

Is it known any closed-form function of the infinite sum of the normalized probabilist's Hermite polynomials, i.e., \begin{align} \sum_{j=0}^\infty \frac{He_j(t)}{\sqrt{j!}} \end{align} where $He_j(t)$ is the probabilist's Hermite polynomial of degree $j \ge 0$. This was previously asked in this thread but not solved yet. Numerically, it seems to be converged to some deterministic function, but still unclear whether it indeed converges. Any comment or suggestion would be greatly appreciated. Thanks!