New series representations for Apéry's and other classical constants (Q1714984)

The authors establish an identity involving the sums \(A_k(a,p,N) =\sum_{j=0}^{k-1} \left((-1)^{jN}/(a+j)^p\right)\) and use it to produce new series representations for \(\log(2)\), \(\zeta(3)\), \(\pi^2\) and the constant of Catalan. For example they obtain \[ \sum_{k=1}^\infty \frac{1}{k^3} = \zeta(3) = \frac{149}{144} + \frac{1}{8} \sum_{k=2}^\infty \frac{(2k+1)(k^4+2k^3+3k^2+2k-2)}{((k-1)k(k+1)(k+2))^2} \left(2H_k^{(2)}-H_{[k/2]}^{(2)}\right), \] where \(H_k^{(2)} = \sum_{j=1}^k1/j^2\).