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

Abstract

In a recent work, Dancs and He found an Euler-type formula for $\zeta (2n+1)$, $n$ being a positive integer, which contains a series they could not reduce to a finite closed-form. This open problem reveals a greater complexity in comparison to $\zeta (2n)$, which is a rational multiple of ${\pi }^{2n}$. For the Dirichlet beta function, the things are `inverse': $\beta (2n+1)$ is a rational multiple of ${\pi }^{2n+1}$ and no closed-form expression is known for $\beta (2n)$. Here in this work, I modify the Dancs-He approach in order to derive an Euler-type formula for $\beta (2n)$, including $\beta (2)=G$, the Catalan's constant. I also convert the resulting series into zeta series, which yields new exact closed-form expressions for a class of zeta series involving $\beta (2n)$ and a finite number of odd zeta values. A closed-form expression for a certain zeta series is also conjectured.

11 pages, no figures. A few small corrections. ACCEPTED for publication in: Integral Transf. Special Functions (09/11/2011)

References in corpus (1)

• Notes on a paper of Tyagi and Holm: A new integral representation for the Riemann Zeta function

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• A simpler proof of a Katsurada's theorem and rapidly converging series for $\zeta (2n+1)$ and $\beta (2n)$
• On the asymptotic behavior of unimodal rank generating functions