Hyperbolic-sine analogues of Eisenstein series, generalized Hurwitz numbers, and $q$-zeta functions

TL;DR

This paper introduces hyperbolic-sine analogues of Eisenstein series and generalized Hurwitz numbers, providing new evaluations of double series and $q$-zeta functions, extending classical results of Hurwitz, Ramanujan, and others.

Contribution

It generalizes classical formulas involving Eisenstein series and Dirichlet series with hyperbolic functions through new double series and generalized Hurwitz numbers.

Findings

Derived new evaluation formulas for double Eisenstein-type series.

Extended classical results to hyperbolic-sine analogues and $q$-zeta functions.

Connected recent formulas to historical results of Hurwitz and Ramanujan.

Abstract

We consider certain double series of Eisenstein type involving hyperbolic-sine functions. We define certain generalized Hurwitz numbers, in terms of which we evaluate those double series. Our main results can be regarded as a certain generalization of well-known results of Hurwitz, Herglotz, Katayama and so on. Our results also include recent formulas of the third-named author which are double analogues of the formulas of Cauchy, Mellin, Ramanujan, Berndt and so on, about certain Dirichlet series involving hyperbolic functions. As an application, we give some evaluation formulas for $q$-zeta functions at positive integers.