I've encountered: $$\Phi(z, s, \alpha) = \sum_{k=0}^\infty \frac { z^k} {(k+\alpha)^s}.$$
When trying to compute:
$$\frac{1}{x}\sum_{p=0}^m \frac{2}{(2p-1)\ x^{2p-1}}\ s.t. x\in\mathbb{N} =\ ???$$ which (by Mathematica): $$???\ =\ \frac{2}{x}\tanh^{-1}(\frac{1}{x}) - \frac{\Phi(\frac{1}{x^2}, 1, m+\frac{1}{2})}{x^{2(m+1)}}$$
Unfortunately, $$ \frac{\Phi(\frac{1}{x^2}, 1, m+\frac{1}{2})}{x^{2(m+1)}} = \frac {1}{x^{2(m+1)}}\sum_{k=0}^\infty \frac {1} {(k+m+\frac{1}{2})x^{2k}}$$
Which doesn't quite help me. Need an efficient way to calculate/estimate this, ideally a convergent formula.
Please see pages 1-5 of the following resource: https://arxiv.org/pdf/1806.01122.pdf
From the following question, Summing Lerch Transcendents, for possible leads.