This problem arose in finding a sum of expectations of Geometric RVs.
Any idea how to simplify the sum
$$\sum_{k=0}^{\frac{n}{2}-1}\frac{1}{(2n^2)^{c'}-(n^2+4k^2)^{c'}}$$
where $c'$ is an integer. It would be good to obtain some digamma function from it.