A friend of mine asked me if I could find a closed form for the series: $$ S = \sum_{n=-\infty}^{\infty} (n-h)^{\alpha} e^{-\beta(n-h)^2}, $$ with $\alpha,\beta > 0$.
I don't even know how to tackle the problem TBH, but it caught my attention and I'd like to know how to solve it.
If $~h$ is and integer and a is an even integer, then the series can be expressed in terms of the
derivative of order $\dfrac a2$ with regard to b of the Jacobi $\theta$ function $~\theta_3\Big(0,e^{-b}\Big),~$ where $\theta_3(0,x)$
$=\displaystyle\sum_{n=-\infty}^\infty x^{n^2}$