I've this series:
$$ \sum_{\ell = 1}^{+ \infty} e^{-t \ \ell^2} \sin{(k\ell)} = f(k, t) $$
where $ t \in [0,\infty]$ , $ k \in [0,2\pi] $.
I need the limit of series like an analytic function of $ k $ and $ \ell $ : $ f( k , \ell)$ .
I tried to transform the sine to exponential but without result. This series is quite similar to the third theta Jacobi elliptic function.
For details check this link: http://people.math.sfu.ca/~cbm/aands/page_567.htm pag $\sim $ 575
My question is: how can I sum that series? Is there some good method to get a result (no matter in terms of what kind of functions)?
