Find the sum of series

Question

Does there exist an explicit formula for the sum of the series $$ \sum_{n=1}^\infty \frac{1}{n^2-z^2}? $$

Answer 1

Equation $(18)$ on this page states that $$\pi \cot(\pi z)=\frac{1}{z}+2z\sum_{n=1}^\infty \frac{1}{z^2-n^2}.$$

Answer 2

This answer handles precisely this question by showing that the Cauchy Principal Value of $$ \sum_{k\in\mathbb{Z}}\frac1{k+z}=\pi\cot(\pi z) $$ and derives an explicit value for $\displaystyle\sum_{k=1}^\infty\frac{1}{k^2-z^2}$ in $(9)$.

Answer 3

Yes, there does. Of course the sum is not defined if $z$ is a nonzero integer.