Generalized rational zeta series for $\zeta (2n)$ and $\zeta (2n+1)$

Abstract

In this paper, we find rational zeta series with $\zeta (2n)$ in terms of $\zeta (2k+1)$ and $\beta (2k)$, the Dirichlet beta function. We then develop a certain family of generalized rational zeta series using the generalized Clausen function and use those results to discover a second family of generalized rational zeta series. As a special case of our results from Theorem 3.1, we prove a conjecture given in 2012 by F.M.S. Lima. Later, we use the same analysis but for the digamma function $\psi (x)$ and negapolygammas ${\psi }^{(-m)}(x)$. With these, we extract the same two families of generalized rational zeta series with $\zeta (2n+1)$ on the numerator rather than $\zeta (2n)$.

27 pages

Cited by in corpus (2)

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