I would be interested in finding a closed form for the following series:
$$f(t):=\sum_{n=1}^\infty \text{csch}^2\left (\frac{2n\pi^2}{t} \right )$$
For come complex $t \ne 0$.
I have tried to use the main definition of Hyperbolic Cosecant, and then making the change of variables $k=e^{\frac{2\pi^2}{t}}$:
$$f(t)=4\sum_{n=1}^\infty \frac{1}{\left (k^{n}-k^{-n}\right )^2}$$
Using this definition, it is easier to test the convergence of the series. However, I do not know how to find a closed form nor how to continue working with the function in this way.
By another change of variables, we could get something similar to:
$$f(t) = -\sum_{n=1}^\infty \frac{1}{\sin^2(nz)}$$
Which seems to look like a more common/famous series.
Any help/hint/happy idea would be welcome.
Thank you.