Bilateral zeta functions associated with the multiple sine functions

TL;DR

This paper introduces two types of bilateral zeta functions linked to multiple sine functions, establishing their key properties and enabling generalizations and simplified constructions within multiple sine function theory.

Contribution

It presents the first definitions of bilateral zeta functions associated with multiple sine functions and explores their fundamental properties and applications.

Findings

Derived Fourier expansions and analytic continuations

Established differential and difference equations

Provided generalizations of multiple sine functions

Abstract

We introduce two types bilateral zeta functions, which are related to the primitive and normalized multiple sine functions respectively. Further, we establish their main properties, that is, Fourier expansions, analytic continuations, differential and difference equations, special values. By applying these results, we obtain not only some generalization of the primitive and normalized multiple sine functions but also simple construction of the multiple sine function theory.