There exists a way to expand a gaussian function into a series of Hermite polynomials as
$$ \sum_{n=0}^{\infty}r^{n}\left[H_{n}(x)\right]^{2} = \frac{1}{\sqrt{\pi(1-r^{2})}}\exp\left(\frac{2r}{1+r}x^{2}\right) $$
see this answer. There also exists a more general formula,
$$ \sum_{n=0}^\infty \frac{H_n(x) H_n(y)}{n!} \left(\frac u 2\right)^n = \frac{1}{\sqrt{1 - u^2}} e^{\frac{2u}{1 + u}xy - \frac{u^2}{1 - u^2}(x - y)^2} $$
see Wikipedia.
My question is whether there exists a generalization of this sum onto the complex numbers, that is for complex valued $u$. If possible, please give references to helpful literature.