I have been working on a problem in Quantum Field Theory that involves some sums that I haven't been able to get a closed form. They all involve hyperbolic functions:
\begin{gather} \sum_{m=1}^{\infty}\frac{1}{m^3} \coth \left(\frac{m\pi}{2x}\right)\,,\nonumber\\ \sum_{m=1}^{\infty}\frac{1}{m^2} \operatorname{csch} ^{2}\left(\frac{m\pi}{2x}\right)\,,\\ \sum_{m=1}^{\infty}\frac{1}{m} \coth \left(\frac{m\pi}{2x}\right) \operatorname{csch}^{2}\left(\frac{m\pi}{2x}\right)\nonumber\,. \end{gather}
I have wondered if anyone knows how to evaluate them and could help me out.
Any references that may address the computation of them are very welcome, too.
Not An Answer. Such series probably have no explicit closed-forms for arbitrary $x$. But for $x=1/2$, we have \begin{aligned} &\sum_{n=1}^{\infty} \frac{\coth(\pi n)}{n^3} =\frac{7\pi^3}{180},\\ &\sum_{n=1}^{\infty} \frac{1}{n^2\sinh(\pi n)^2} =\frac23G-\frac{11\pi^2}{180},\\ &\sum_{n=1}^{\infty} \frac{\coth(\pi n)}{n\sinh(\pi n)^2}=\frac\pi{30}-\frac{G}{3\pi}. \end{aligned} where $G=\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)^2}$ denotes Catalan's constant.
Proof. The first sum is $$ \sum_{m,n\ge1} \frac{1}{n^2(m^2+n^2)}=\sum_{n=1}^{\infty} \frac{n\pi\coth(\pi n)-1}{2n^4} =\frac{1}{2} \sum_{m,n\ge1}\frac{m^2+n^2}{m^2n^2(m^2+n^2)} =\frac{\pi^4}{72}. $$ For the last two sums, explicitly compute $\sum_{m\in\mathbb{Z},n\ge1}\frac{1}{(m^2+n^2)^2} = \frac{\pi^2}{3}G-\frac{\pi^4}{90} , \sum_{m\in\mathbb{Z},n\ge1}\frac{n^2}{(m^2+n^2)^3} = \frac{\pi^2}{6}G ,$ and it gives the claimed results.