TITLE: Calculating in closed form another digamma alternating series
QUESTION [5 upvotes]: Is there any clever way of finish it fastly?
$$\sum _{n=1}^{\infty } (-1)^{n+1} \left(\psi ^{(0)}\left(\frac{5}{8}+\frac{3 n}{8}\right)-\psi ^{(0)}\left(\frac{1}{8}+\frac{3 n}{8}\right)\right)$$
Maybe using series representation of digamma or other ways you might like to share. Or is it possible to finish it by series manipulation only (as a distinct matter I'm interested in)?
Supplementary question: prove that 
$$\sum _{n=1}^{\infty } (-1)^{n+1} \left(\psi ^{(0)}\left(\frac{5}{8}+\frac{9 n}{32}\right)-\psi ^{(0)}\left(\frac{1}{8}+\frac{9 n}{32}\right)\right)$$
$$=\frac{64}{9} \sin \left(\frac{\pi }{9}\right) \tan ^{-1}\left(\cot \left(\frac{\pi }{9}\right)-\csc \left(\frac{\pi }{9}\right)\right)-\frac{32 \tan ^{-1}\left(\cot \left(\frac{\pi }{9}\right)-\csc \left(\frac{\pi }{9}\right)\right)}{3 \sqrt{3}}-2 \tan ^{-1}\left(\cot \left(\frac{\pi }{16}\right)-\csc \left(\frac{\pi }{16}\right)\right)-2 \tan ^{-1}\left(\cot \left(\frac{\pi }{16}\right)+\csc \left(\frac{\pi }{16}\right)\right)-2 \tan ^{-1}\left(\left(1-\cos \left(\frac{3 \pi }{16}\right)\right) \csc \left(\frac{3 \pi }{16}\right)\right)+2 \tan ^{-1}\left(\cot \left(\frac{3 \pi }{16}\right)+\csc \left(\frac{3 \pi }{16}\right)\right)-2 \tan ^{-1}\left(\sec \left(\frac{\pi }{16}\right)-\tan \left(\frac{\pi }{16}\right)\right)+2 \tan ^{-1}\left(\sec \left(\frac{\pi }{16}\right)+\tan \left(\frac{\pi }{16}\right)\right)+2 \tan ^{-1}\left(\sec \left(\frac{3 \pi }{16}\right)-\tan \left(\frac{3 \pi }{16}\right)\right)-2 \tan ^{-1}\left(\sec \left(\frac{3 \pi }{16}\right)+\tan \left(\frac{3 \pi }{16}\right)\right)+4 \tan ^{-1}\left(\cot \left(\frac{\pi }{16}\right)-\csc \left(\frac{\pi }{16}\right)\right) \cos \left(\frac{\pi }{8}\right)+4 \tan ^{-1}\left(\cot \left(\frac{\pi }{16}\right)+\csc \left(\frac{\pi }{16}\right)\right) \cos \left(\frac{\pi }{8}\right)-4 \tan ^{-1}\left(\sec \left(\frac{\pi }{16}\right)-\tan \left(\frac{\pi }{16}\right)\right) \cos \left(\frac{\pi }{8}\right)+4 \tan ^{-1}\left(\sec \left(\frac{\pi }{16}\right)+\tan \left(\frac{\pi }{16}\right)\right) \cos \left(\frac{\pi }{8}\right)-4 \tan ^{-1}\left(\left(1-\cos \left(\frac{3 \pi }{16}\right)\right) \csc \left(\frac{3 \pi }{16}\right)\right) \cos \left(\frac{\pi }{16}\right)-4 \tan ^{-1}\left(\cot \left(\frac{3 \pi }{16}\right)+\csc \left(\frac{3 \pi }{16}\right)\right) \cos \left(\frac{\pi }{16}\right)+4 \tan ^{-1}\left(\sec \left(\frac{\pi }{16}\right)-\tan \left(\frac{\pi }{16}\right)\right) \cos \left(\frac{\pi }{16}\right)+4 \tan ^{-1}\left(\sec \left(\frac{\pi }{16}\right)+\tan \left(\frac{\pi }{16}\right)\right) \cos \left(\frac{\pi }{16}\right)+4 \tan ^{-1}\left(\sec \left(\frac{\pi }{16}\right)-\tan \left(\frac{\pi }{16}\right)\right) \cos \left(\frac{3 \pi }{16}\right)+4 \tan ^{-1}\left(\sec \left(\frac{\pi }{16}\right)+\tan \left(\frac{\pi }{16}\right)\right) \cos \left(\frac{3 \pi }{16}\right)+4 \tan ^{-1}\left(\sec \left(\frac{3 \pi }{16}\right)-\tan \left(\frac{3 \pi }{16}\right)\right) \cos \left(\frac{3 \pi }{16}\right)+4 \tan ^{-1}\left(\sec \left(\frac{3 \pi }{16}\right)+\tan \left(\frac{3 \pi }{16}\right)\right) \cos \left(\frac{3 \pi }{16}\right)-8 \pi  \cos \left(\frac{3 \pi }{16}\right)-\frac{64}{9} \tan ^{-1}\left(\sec \left(\frac{\pi }{18}\right)+\tan \left(\frac{\pi }{18}\right)\right) \cos \left(\frac{\pi }{18}\right)+\frac{32}{9} \pi  \cos \left(\frac{\pi }{18}\right)+\frac{32}{9} \cos \left(\frac{2 \pi }{9}\right) \log (2)+\frac{7 \log (2)}{9}-\frac{32}{9} \cos \left(\frac{2 \pi }{9}\right) \log \left(2-2 \cos \left(\frac{\pi }{9}\right)\right)-\frac{32}{9} \cos \left(\frac{\pi }{9}\right) \log \left(\cos \left(\frac{2 \pi }{9}\right)+1\right)+2 \cos \left(\frac{\pi }{8}\right) \log \left(\cos \left(\frac{\pi }{16}\right)+1\right)-\sqrt{2} \log \left(\cos \left(\frac{\pi }{16}\right)+1\right)+2 \cos \left(\frac{\pi }{8}\right) \log \left(2-2 \cos \left(\frac{\pi }{16}\right)\right)-\sqrt{2} \log \left(2-2 \cos \left(\frac{\pi }{16}\right)\right)+2 \cos \left(\frac{\pi }{16}\right) \log \left(\cos \left(\frac{3 \pi }{16}\right)+1\right)+\sqrt{2} \log \left(\cos \left(\frac{3 \pi }{16}\right)+1\right)-2 \cos \left(\frac{\pi }{16}\right) \log \left(2-2 \cos \left(\frac{3 \pi }{16}\right)\right)+\sqrt{2} \log \left(2-2 \cos \left(\frac{3 \pi }{16}\right)\right)-2 \cos \left(\frac{\pi }{8}\right) \log \left(\sin \left(\frac{\pi }{16}\right)+1\right)-2 \cos \left(\frac{\pi }{16}\right) \log \left(\sin \left(\frac{\pi }{16}\right)+1\right)-2 \cos \left(\frac{3 \pi }{16}\right) \log \left(\sin \left(\frac{\pi }{16}\right)+1\right)-\sqrt{2} \log \left(\sin \left(\frac{\pi }{16}\right)+1\right)-2 \cos \left(\frac{\pi }{8}\right) \log \left(2-2 \sin \left(\frac{\pi }{16}\right)\right)+2 \cos \left(\frac{\pi }{16}\right) \log \left(2-2 \sin \left(\frac{\pi }{16}\right)\right)+2 \cos \left(\frac{3 \pi }{16}\right) \log \left(2-2 \sin \left(\frac{\pi }{16}\right)\right)-\sqrt{2} \log \left(2-2 \sin \left(\frac{\pi }{16}\right)\right)+2 \cos \left(\frac{3 \pi }{16}\right) \log \left(\sin \left(\frac{3 \pi }{16}\right)+1\right)+\sqrt{2} \log \left(\sin \left(\frac{3 \pi }{16}\right)+1\right)-2 \cos \left(\frac{3 \pi }{16}\right) \log \left(2-2 \sin \left(\frac{3 \pi }{16}\right)\right)+\sqrt{2} \log \left(2-2 \sin \left(\frac{3 \pi }{16}\right)\right)+\frac{32}{9} \cos \left(\frac{\pi }{9}\right) \log \left(\sin \left(\frac{\pi }{18}\right)+1\right)+\frac{32}{9} \cos \left(\frac{2 \pi }{9}\right) \log \left(\sin \left(\frac{\pi }{18}\right)+1\right)+4 \tan ^{-1}\left(\left(1-\cos \left(\frac{3 \pi }{16}\right)\right) \csc \left(\frac{3 \pi }{16}\right)\right) \sin \left(\frac{\pi }{8}\right)-4 \tan ^{-1}\left(\cot \left(\frac{3 \pi }{16}\right)+\csc \left(\frac{3 \pi }{16}\right)\right) \sin \left(\frac{\pi }{8}\right)+4 \tan ^{-1}\left(\sec \left(\frac{3 \pi }{16}\right)-\tan \left(\frac{3 \pi }{16}\right)\right) \sin \left(\frac{\pi }{8}\right)-4 \tan ^{-1}\left(\sec \left(\frac{3 \pi }{16}\right)+\tan \left(\frac{3 \pi }{16}\right)\right) \sin \left(\frac{\pi }{8}\right)+2 \log \left(\cos \left(\frac{3 \pi }{16}\right)+1\right) \sin \left(\frac{\pi }{8}\right)+2 \log \left(2-2 \cos \left(\frac{3 \pi }{16}\right)\right) \sin \left(\frac{\pi }{8}\right)-2 \log \left(\sin \left(\frac{3 \pi }{16}\right)+1\right) \sin \left(\frac{\pi }{8}\right)-2 \log \left(2-2 \sin \left(\frac{3 \pi }{16}\right)\right) \sin \left(\frac{\pi }{8}\right)+\frac{32}{9} \pi  \sin \left(\frac{\pi }{9}\right)+\frac{64}{81} \pi  \sin \left(\frac{2 \pi }{9}\right)-4 \tan ^{-1}\left(\cot \left(\frac{\pi }{16}\right)-\csc \left(\frac{\pi }{16}\right)\right) \sin \left(\frac{\pi }{16}\right)+4 \tan ^{-1}\left(\cot \left(\frac{\pi }{16}\right)+\csc \left(\frac{\pi }{16}\right)\right) \sin \left(\frac{\pi }{16}\right)-4 \tan ^{-1}\left(\sec \left(\frac{3 \pi }{16}\right)-\tan \left(\frac{3 \pi }{16}\right)\right) \sin \left(\frac{\pi }{16}\right)-4 \tan ^{-1}\left(\sec \left(\frac{3 \pi }{16}\right)+\tan \left(\frac{3 \pi }{16}\right)\right) \sin \left(\frac{\pi }{16}\right)+2 \log \left(\cos \left(\frac{\pi }{16}\right)+1\right) \sin \left(\frac{\pi }{16}\right)-2 \log \left(2-2 \cos \left(\frac{\pi }{16}\right)\right) \sin \left(\frac{\pi }{16}\right)-2 \log \left(\sin \left(\frac{3 \pi }{16}\right)+1\right) \sin \left(\frac{\pi }{16}\right)+2 \log \left(2-2 \sin \left(\frac{3 \pi }{16}\right)\right) \sin \left(\frac{\pi }{16}\right)-4 \tan ^{-1}\left(\left(1-\cos \left(\frac{\pi }{16}\right)\right) \csc \left(\frac{\pi }{16}\right)\right) \sin \left(\frac{3 \pi }{16}\right)-4 \tan ^{-1}\left(\cot \left(\frac{\pi }{16}\right)+\csc \left(\frac{\pi }{16}\right)\right) \sin \left(\frac{3 \pi }{16}\right)+4 \tan ^{-1}\left(\left(1-\cos \left(\frac{3 \pi }{16}\right)\right) \csc \left(\frac{3 \pi }{16}\right)\right) \sin \left(\frac{3 \pi }{16}\right)+4 \tan ^{-1}\left(\cot \left(\frac{3 \pi }{16}\right)+\csc \left(\frac{3 \pi }{16}\right)\right) \sin \left(\frac{3 \pi }{16}\right)-4 \log (2) \sin \left(\frac{3 \pi }{16}\right)-2 \log \left(\cos \left(\frac{\pi }{16}\right)+1\right) \sin \left(\frac{3 \pi }{16}\right)+2 \log \left(2-2 \cos \left(\frac{\pi }{16}\right)\right) \sin \left(\frac{3 \pi }{16}\right)-2 \log \left(\cos \left(\frac{3 \pi }{16}\right)+1\right) \sin \left(\frac{3 \pi }{16}\right)+2 \log \left(2-2 \cos \left(\frac{3 \pi }{16}\right)\right) \sin \left(\frac{3 \pi }{16}\right)-\frac{32}{9} \log (2) \sin \left(\frac{\pi }{18}\right)+\frac{32}{9} \log \left(2-2 \cos \left(\frac{\pi }{9}\right)\right) \sin \left(\frac{\pi }{18}\right)-\frac{32}{9} \log \left(\cos \left(\frac{2 \pi }{9}\right)+1\right) \sin \left(\frac{\pi }{18}\right)-\frac{32 \tan ^{-1}\left(\sec \left(\frac{\pi }{18}\right)+\tan \left(\frac{\pi }{18}\right)\right)}{3 \sqrt{3}}+\frac{64 \pi }{27 \sqrt{3}}$$

REPLY [3 votes]: Using the representation
$$\psi(x) = \int_0^{\infty} dt \left (\frac{e^{-t}}{t} - \frac{e^{-x t}}{1-e^{-t}} \right ) $$
and changing the order of summation and integration, we get the following integral representation of the sum:
$$\int_0^{\infty} dt \frac{e^{-t/2}}{\left ( 1+e^{-t/2} \right )\left ( 1+e^{-3 t/8} \right )} $$
Subbing $u=e^{-t/8}$, we get
$$8 \int_0^1 du \frac{u^3}{(1+u^3)(1+u^4)} $$