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

Abstract

In a recent work, Dancs and He found an Euler-type formula for $\,ζ{(2\,n+1)}$, $\,n\,$ 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 $ζ(2n)$, which is a rational multiple of $π^{2n}$. For the Dirichlet beta function, the things are `inverse': $β(2n+1)$ is a rational multiple of $π^{2n+1}$ and no closed-form expression is known for $β(2n)$. Here in this work, I modify the Dancs-He approach in order to derive an Euler-type formula for $\,β{(2n)}$, including $\,β{(2)} = G$, 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 $\,β{(2n)}$ and a finite number of odd zeta values. A closed-form expression for a certain zeta series is also conjectured.