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

TL;DR

This paper derives an Euler-type formula for the Dirichlet beta function at even integers, including Catalan's constant, and provides new closed-form expressions for related zeta series, advancing understanding of these special functions.

Contribution

It introduces a modified approach to obtain an Euler-type formula for beta(2n), including new closed-form expressions for zeta series involving beta(2n) and odd zeta values.

Findings

Derived Euler-type formula for beta(2n) including beta(2) = G

Converted series into zeta series for closed-form expressions

Proposed a conjecture for a specific zeta series

Abstract

In a recent work, Dancs and He found an Euler-type formula for $\,\zeta{(2\,n+1)}$, $\,n\,$ 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.