Barnes multiple zeta-functions, Ramanujan's formula, and relevant series involving hyperbolic functions

TL;DR

This paper explores functional equations of Barnes multiple zeta-functions, deriving Ramanujan's formula from them, and generalizes series involving hyperbolic functions using Bernoulli polynomials, extending classical results.

Contribution

It introduces new functional equations for Barnes zeta-functions and generalizes classical hyperbolic series formulas with multiple versions.

Findings

Ramanujan's formula derived from Barnes zeta-function equations

Generalized hyperbolic series formulas using Bernoulli polynomials

Provided multiple new versions of classical series formulas

Abstract

In the former part of this paper, we give functional equations for Barnes multiple zeta-functions and consider some relevant results. In particular, we show that Ramanujan's classical formula for the Riemann zeta values can be derived from functional equations for Barnes zeta-functions. In the latter half part, we generalize some evaluation formulas of certain series involving hyperbolic functions in terms of Bernoulli polynomials. The original formulas were classically given by Cauchy, Mellin, Ramanujan, and later recovered and formulated by Berndt. From our consideration, we give multiple versions of these known formulas.