A BBP-type series

Question

The BBP-type series

$$ \frac{\pi}{2} \, \left( \frac{\alpha^{2}}{5} \right)^{\frac{1}{4}} = \sum_{n=0}^{\infty} \left[ \frac{1}{10 n + 1} + \frac{\alpha}{10 n + 3} - \frac{\alpha}{10 n + 7} - \frac{1}{10 n + 9} \right],$$

with golden ratio $\alpha = \frac{1 + \sqrt{5}}{2}$ can be obtained by a particular Sine series. The questions proposed here are:

1. Can multiple Fourier-Sine/Cosine series yield the same result?
2. Are there non-Fourier series methods which yield this series?

Answer 1

Hint. One may recall the following series representation of the digamma function, $$ \sum_{k=1}^{\infty} \left( \frac{1}{k} - \frac{1}{u+k} \right)=\psi(u+1)+\gamma, \qquad u >-1, $$ giving $$ \sum_{k=0}^{\infty} \left( \frac{1}{k+a} - \frac{1}{k+b} \right)=\psi(b)-\psi(a),\qquad a>0,\,b>0. \tag1 $$

From $(1)$ one gets $$ \begin{align} &10\cdot\sum _{n=0}^{\infty } \left(\frac{1}{10 n+a}+\frac{\alpha }{10 n+b}-\frac{\alpha }{10 n+c}-\frac{1}{10 n+d}\right) \\\\&=-\psi\left(\frac{a}{10}\right)-\alpha\: \psi\left(\frac{b}{10}\right)+\alpha\: \psi\left(\frac{c}{10}\right)+\psi\left(\frac{d}{10}\right) \end{align} $$ then, putting $$ a=1,\quad b=3,\quad c=7,\quad d=9, $$ using Gauss' digamma theorem one deduces

$$ \begin{align} 10\cdot\sum_{n=0}^{\infty} \left( \frac{1}{10 n + 1} + \frac{\alpha}{10 n + 3} - \frac{\alpha}{10 n + 7} - \frac{1}{10 n + 9} \right) =\left(\sqrt{5+2\sqrt{5}}+\sqrt{5-2\sqrt{5}}\:\alpha\right)\pi \end{align} $$

from which one obtains the announced result.