The temperature Green's function in momentum-frequency representation of a system of free phonons (in one-dimension) is given by [1]:
\begin{equation} D^{(0)}(k_n,\omega_{n'}) = -m\frac{\omega_{k_n}^2}{\omega_{n'}^2+\omega_{k_n}^2}, \end{equation} where $\omega_k$ is the phonon dispersion, \begin{equation} \omega_{k_n}^2 = 4\frac{\kappa}{m}\sin^2\frac{k_n}{2}. \end{equation} In the context of calculating the temperature Green's function for a toy (interacting) problem, I need the full perturbation series for the exact Green's function in position-frequency space. I therefore have to compute discrete inverse Fourier transforms which amount to finite sums of the following form:
\begin{eqnarray} D^{(0)}_{j0}(\omega_{n'}) &=& -m\sum_{k_n=0}^{\pi} \frac{\sin^2\frac{k_n}{2}}{a^2+\sin^2\frac{k_n}{2}}e^{ik_nj} \\ &=& -m\sum_{n=0}^{N/2} \frac{\sin^2\frac{n\pi}{N}}{a^2+\sin^2\frac{n\pi}{N}}e^{i \frac{2\pi n}{N} j}, \end{eqnarray} where $k_n=\frac{2\pi n}{N}$ with $0\le n\le N/2$, and $j$ is the position coordinate (the model is on a lattice so this labels the site), and $a$ is a constant. $N$ is even.
Any advice on the possibility of computing the sums of the type above would be greatly appreciated!
[1] Abrikosov, A. A.; Gorkov, L. P.; Dzyaloshinski, I. E., Methods of quantum field theory in statistical physics, Englewood Cliffs, N.J.: Prentice-Hall, Inc. XV, 352 p. (1963). ZBL0135.45003.: