Infinite sum of gaussian exponential

Question

Does anybody know a closed expression for: $$\sum_{n=-\infty}^{+\infty} e^{-(a+bn)^2}$$

Answer 1

Jacobi theta function: $\theta(z,\tau)=\sum\limits_{n=-\infty}^{+\infty} e^{i\pi n^2\tau+2i\pi n z}$

So $\sum\limits_{n=-\infty}^{+\infty} e^{-(a+bn)^2}=e^{-a^2}\theta(i\frac{ab}{\pi},i\frac{b^2}{\pi})$