Generating function of $\coth(n)$

Question

Is there a generating function known for $\coth(kn)$? where $k$ is a constant. That is

$$\sum_{n\geqslant1}\coth(kn)x^n$$

If yes, then can you please provide a reference? Thanks for reading.

Answer 1

Assume that $k>0$. Then $$ \coth (kn) = 1 + 2\sum\limits_{m = 1}^\infty {\mathrm{e}^{ - 2mkn} } . $$ Thus, if $\left| x \right| < 1$, $$ \sum\limits_{n = 1}^\infty {\coth (kn)x^n } =\frac{x}{{1 - x}} + 2x\sum\limits_{m = 1}^\infty {\frac{1}{{\mathrm{e}^{2mk} - x}}} . $$ You can express the right-hand side in terms of the $q$-digamma function if you like. Indeed $$ 2x\sum\limits_{m = 1}^\infty {\frac{1}{{\mathrm{e}^{2mk} - x}}}=\frac{{\psi _{\mathrm{e}^{2k} }\big( {1 - \frac{{\log x}}{{2k}}} \big) + \log x - k + \log (\mathrm{e}^{2k} - 1)}}{{k}}. $$