Evaluate $\sum\limits_{n=0}^\infty n^ke^{-an^2}$ and $\sum\limits_{n=0}^\infty(-1)^nn^ke^{-an^2}$ .

Question

Evaluate $\sum\limits_{n=0}^\infty n^ke^{-an^2}$ and $\sum\limits_{n=0}^\infty(-1)^nn^ke^{-an^2}$ .

I know for the special case $k=0$ , they relate to Jacobi Theta Functions.

Answer 1

$\sum\limits_{n=0}^\infty(-1)^nn^ke^{-an^2}=2^{k+1}\sum\limits_{n=0}^\infty n^ke^{-4an^2}-\sum\limits_{n=0}^\infty n^ke^{-an^2}$

Therefore it's enough to compute $\sum\limits_{n=0}^\infty n^ke^{-an^2}$.

Answer 2

A hint to compute $\sum\limits_{n=0}^\infty n^{k}e^{-an^2}$ for even $k$:
$$\sum\limits_{n=0}^\infty n^{2m}e^{-an^2} =\left(-\frac{\partial}{\partial a}\right)^m \sum\limits_{n=0}^\infty e^{-an^2}$$