Summing the following two series I have encountered the following two sums:
\begin{equation}
S_1(\alpha, N) = \sum\limits_{j=1}^{N-1} \dfrac{e^{i\alpha \sin(\phi_j/2)}}{\sin^k(\phi_j/2)}, \quad S_2(\alpha, N) = \sum\limits_{j=1}^{N-1} \dfrac{e^{i\alpha \sin(\phi_j/2)} \cos(\phi_j)}{\sin^k(\phi_j/2)},
\end{equation}
where $1\le k\le3$, and $\phi_j = \frac{2 \pi j }{N}$.
I tried to replace sums with integrals, but in this case a change of variable is required to $x=\sin(\phi_j/2)$ or $x=\cos(\phi_j/2)$, and things like $\sqrt{1-x^2}$ appear, which is nasty to have.
Any good ideas of how to calculate analytically these types of sums? (Though I have no guarantee that it is possible to do precisely, even some approximations will be good!)
 A: Using some physical rationale (these series appear in a physical problem), I was able to obtain a fairly good approximation for specific linear combinations of the sums above in the limit $N \to \infty$. I am too lazy to prove this mathematically, so I will give away the answer and the numerical check. Let us define the $6$ following sums:
\begin{eqnarray}
\Sigma_1(\alpha, N) = \dfrac{1}{4} \sum\limits_{j=1}^{N-1} \dfrac{1+\cos(\phi_j)}{\alpha \sin(\phi_j/2)} e^{i \alpha \sin(\phi_j/2)}, \quad \tilde{\Sigma}_1(\alpha, N) = \dfrac{\rm{Li}_1(e^{i \alpha \sin(\phi_1/2)})}{\alpha \sin(\phi_1/2)}; \nonumber\\
\Sigma_2(\alpha, N) = \dfrac{1}{4} \sum\limits_{j=1}^{N-1} \dfrac{3-\cos(\phi_j)}{(\alpha \sin(\phi_j/2))^2} e^{i \alpha \sin(\phi_j/2)}, \quad \tilde{\Sigma}_2(\alpha, N) = \dfrac{\rm{Li}_2(e^{i \alpha \sin(\phi_1/2)})}{(\alpha \sin(\phi_1/2))^2}; \nonumber\\
\Sigma_3(\alpha, N) = \dfrac{1}{4} \sum\limits_{j=1}^{N-1} \dfrac{-3+\cos(\phi_j)}{(\alpha \sin(\phi_j/2))^3} e^{i \alpha \sin(\phi_j/2)}, \quad \tilde{\Sigma}_3(\alpha, N) = - \dfrac{\rm{Li}_3(e^{i \alpha \sin(\phi_1/2)})}{(\alpha \sin(\phi_1/2))^3}.
\end{eqnarray}
In the above $\rm{Li}_s(z)$ is a polylogarithm special function.
My claim is that for sufficiently large $N$, $\Sigma_j(\alpha,N)\to\tilde{\Sigma}_j(\alpha, N)$. In order to make it illustrative, let us plot these sums as functions of $\alpha$, for some big $N$.
$\Sigma_1$ for N=100" />
$\Sigma_2$ for N=100" />
$\Sigma_3$ for N=100" />
$\Sigma_1$ for N=10000" />
A couple of comments on these pictures:

*

*Sums $\Sigma_2, \Sigma_3$ converge to their limiting values already for $N\sim 100$ nicely (see figures 2, 3 above).

*The worst sum is $\Sigma_1$ bcoz of $1/(\alpha \sin(\phi_j/2))$ term. One needs to take $N\sim 10^{3}-10^{4}$ to achieve a convincing level of convergence (compare figures 1, and 4 above).

*Sum $\tilde{\Sigma}_1$ diverges each time $\alpha \sin(\phi_1/2) = 2\pi m$, where $m$ is some integer. This condition corresponds to what is called Bragg resonance in physics (one can see a couple of these peaks in figures 1, 4 above). This happens because in this case $\rm{Li}_1(1) = \zeta(1) = \infty$, where $\zeta(z)$ is the Riemann zeta-function.

Thank you for attention!
