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

Abstract

In this paper, we find rational zeta series with $ζ(2n)$ in terms of $ζ(2k+1)$ and $β(2k)$, 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 $ψ(x)$ and negapolygammas $ψ^{(-m)}(x)$. With these, we extract the same two families of generalized rational zeta series with $ζ(2n+1)$ on the numerator rather than $ζ(2n)$.