Infinite sum $\frac{e^{-a n^2}}{n^2+b^2}$

Question

I‌ was wondering if the function \begin{equation} f(a,b) = \sum_{n=1}^\infty \frac{e^{-a n^2}}{n^2 + b^2}, \qquad\qquad (a,b>0) \end{equation} has a representation in terms of known elementary/special functions (like theta functions etc)?

Answer 1

It is not too difficult to see that we have the differential equation $$\frac{\partial}{\partial a}f(a,b)-b^2f(a,b)=\frac12(1-\vartheta_3(e^{-a})).$$ Using the integrating factor $M(a)=e^{-ab^2}$, we have $$\frac{\partial}{\partial t}\left(e^{-tb^2}f(t,b)\right)=\frac12e^{-tb^2}(1-\vartheta_3(e^{-t})),$$ then integrating from $t=0$ to $t=a$, we have $$e^{-ab^2}f(a,b)-f(0,b)=\frac12\int_0^a e^{-tb^2}(1-\vartheta_3(e^{-t}))dt.$$ Then from here, we have $$f(0,b)=\frac{\pi}{2b}\coth\pi b-\frac1{2b^2},$$ so that $$f(a,b)=\frac{e^{ab^2}}{2b^2}\left\{\pi b\coth\pi b-1+b^2\int_0^a e^{-tb^2}(1-\vartheta_3(e^{-t}))dt\right\}.$$