Infinite sum reciprocal hyperbolic sine squared

This is a short article to prove the following result

$\begin{align*} \sum_{n=1}^\infty\frac{\pi^2}{n^2 \sinh^2 (\pi n)}&= \frac{2}{3}G-\frac{11}{30}\zeta(2) \end{align*}$

To this end we will rely on some previous established results, namely:

$\begin{align*} \sum_{m=1}^\infty\sum_{m=1}^\infty \frac{1}{(n^2+m^2)^2}&=\frac{\pi^2}{6}G-\frac{\pi^4}{90} \end{align*}$

$\begin{align*} \sum_{n=1}^\infty \frac{\coth(n \pi )}{n^{3} } =\frac{7\pi^3}{180} \end{align*}$

As a corollary of our goal series we also obtain this nice series

$\begin{align*} \sum_{n=0}^\infty\frac{ \coth^2(z)}{n^2} &=\frac23 G+\frac{19}{30}\zeta(2) \end{align*}$

Click here for the proof of our main series.

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