A simpler proof of a Katsurada's theorem and rapidly converging series for $\zeta (2n+1)$ and $\beta (2n)$

Abstract

In a recent work on Euler-type formulae for even Dirichlet beta values, i.e. $\beta (2n)$, I have derived an exact closed-form expression for a class of zeta series. From this result, I have conjectured closed-form summations for two families of zeta series. Here in this work, I begin by using a known formula by Wilton to prove those conjectures. As example of applications, some special cases are explored, yielding rapidly converging series representations for the Apéry constant, $\zeta (3)$, and the Catalan constant, $G=\beta (2)$. Interestingly, our series for $\zeta (3)$ converges faster than that used by Apéry in his irrationality proof (1978). Also, our series for $G$ converges faster than a celebrated one discovered by Ramanujan (1915). At last, I present a simpler, more direct proof for a recent theorem by Katsurada which generalizes the above results.

10 pages, no figures. In press: Annali di Matematica Pura ed Applicata (2014)

References in corpus (1)

• An Euler-type formula for $\beta (2n)$ and closed-form expressions for a class of zeta series

Cited by in corpus (1)

• Renewal sequences and record chains related to multiple zeta sums