Mathematics – Number Theory
Scientific paper
2009-10-26
Mathematics
Number Theory
11 pages, no figures. A few small corrections. ACCEPTED for publication in: Integral Transf. Special Functions (09/11/2011)
Scientific paper
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.
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