Simplifying $\sum_{ m = - \infty}^{\infty} e^{-am^2/2 + bm}$

Question

Is there any way to simplify an expression like this $\sum_{ m = - \infty}^{\infty} e^{-am^2/2 + bm}$? I know there exist an identity for a similar expression, just integrating, does the same identity still hold for the summation case? If so, how can I argue that the identity still holds. Thanks!

Answer 1

Depends on exactly what you mean by "simplify". The summation is equal to $f(e^{b-a/2},e^{-b-a/2})$ where $f(,)$ is Ramanujan's general theta function. I don't know what identity you are referring to.