In the paper hep-th/0812.2909 page 34-35, there's a sum that I've been trying to do explicitly but I can't find a way. The sum is
$$ \frac{2l}{\pi l! (l-1)!} \sum_{k\in\mathbb{Z}} \sum_{n=0}^{\infty} e^{-i (2n+l+1)t+i k (\phi-t) } \frac{\Gamma(n+|k|+l+1)\Gamma(n+l+1)}{\Gamma(n+|k|+1)\Gamma(n+1)}=\frac{l^{2}/(2^{l}\pi)}{[\cos(t-i\epsilon)-\cos(\phi))]^{l+1}} $$
In page 54 the authors need again to do a similar sum(it's actually the same), where they say you have to use something called "Poisson Ressumation" and "sum over the images". I tried to use what i found about Poisson ressumation, but I don't know how to Fourier transform a quotient between gammas, and even if I did, I understand it would turn the left hand side into another sum over integers.
Thanks for your help!
PS: I'm not a native speaker so forgive me if my English is a bit rusty.
I am the OP, but just to put this to rest I solved this problem some years ago in App. A of https://arxiv.org/abs/1808.10306
Hope it saves time to someone!