Board index Computation of Series Series involving harmonic number and hyperbolic sine functi

## Series involving harmonic number and hyperbolic sine functi

Post your questions related to Computation of Series here.

Moderators: galactus, sos440, zaidalyafey

### Series involving harmonic number and hyperbolic sine functi

Fri Jul 07, 2017 2:09 am

Posts: 105
I deduce the formula that
$$\sum\limits_{n = 1}^\infty {\frac{{\pi {H_n}}}{{n\sinh \left( {n\pi } \right)}}} = \frac{1}{2}\zeta \left( 3 \right) + \pi \sum\limits_{n = 1}^\infty {\frac{1}{{{n^2}\sinh \left( {n\pi } \right)}}} + \frac{{{\pi ^2}}}{2}\sum\limits_{n = 1}^\infty {\frac{{\cosh \left( {n\pi } \right)}}{{n{{\sinh }^2}\left( {n\pi } \right)}}} + \sum\limits_{n,k = 1}^\infty {\frac{{k\pi \coth \left( {k\pi } \right)}}{{n\left( {{n^2} + {k^2}} \right)}}{{\left( { - 1} \right)}^k}}$$
However, the sum on the right hand side
$$\sum\limits_{n,k = 1}^\infty {\frac{{k\pi \coth \left( {k\pi } \right)}}{{n\left( {{n^2} + {k^2}} \right)}}{{\left( { - 1} \right)}^k}}=?$$
have further simplifications?