<s>[INST] A BBP-type series 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:


*

*Can multiple Fourier-Sine/Cosine series yield the same result?

*Are there non-Fourier series methods which yield this series?

 [/INST] 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.
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