Is there any more useful form (integral form, as a delta function, etc.) for the sum $$ \sum_n^{\infty} \frac{e^{-\frac{x^2+y^2}{2}}H_n(x)H_n(y)}{2^nn!\sqrt{n+\gamma}}, $$ or for this similar sum $$ \sum_n^{\infty} \frac{e^{-\frac{x^2+y^2}{2}}H_n(x)H_n(y)}{2^nn!\sqrt{n+\gamma}}e^{iz\sqrt{n+\gamma}}, $$ where $\gamma$ is a real number?
Other than trying to use the completeness relation, $$ \sum_n^{\infty} \frac{e^{-\frac{x^2+y^2}{2}}H_n(x)H_n(y)}{2^nn!}=\sqrt{\pi}\delta(x-y), $$ the closest I have found was given here where it was stated $$ \sum_n^{\infty} \frac{H_n(x)z^n}{n!\sqrt{n+\frac{1}{2}}}=\frac{2}{\sqrt{\pi}}\int_0^{\infty}\mathrm{exp}\left(2xze^{-t^2}-z^2e^{-2t^2}-\frac{t^2}{2}\right)dt. $$ Further explanation of this equation would be useful as well.