I am wondering if there is a simpler form of the following summation involving the (physicists') Hermite polynomials:
$$\sum_{k=0}^{n}\frac{H_k(x)}{(2i)^k},$$
where $i=\sqrt{-1}$ is the imaginary unit. I would love to find a "closed form" solution for this summation, perhaps in terms of a product of one or more Hermite polynomials (or another polynomial that is easily computable in MATLAB). I've checked this website as well as my go-to sources for these kinds of problems (Gradshtein and Ryzhik, and DLMF) without any luck.
An operator form, or symbolic for computation, can be seen as $$\sum_{k=0}^{n} \frac{H_{k}(x)}{(2 \, i)^{k}} = (-i)^{n} \, e^{- D^{2}/4} \, \left( \frac{x^{n+1} - i^{n+1}}{x - i} \right).$$ This is obtained by using $$H_{n}(x) = e^{- D^{2}/4} \, (2 x)^{n}$$ and then summing the resulting series.