Narendra Bhandari

Positive rational series for reciprocal powers of Catalan's constant and Dirichlet beta values

Let $\beta(s)=\sum_{k=0}^{\infty}(-1)^k(2k+1)^{-s}$ and let $G=\beta(2)$ be Catalan's constant. We develop two families of positive series for reciprocal powers $\beta(s)^{-r}$. The first is obtained from the classical Euler transformation and is evaluated at $1/2$; it is valid for real $s>0$. A second transformation,...