Mordell-Tornheim multiple zeta-functions, their integral analogues, and relations among multiple polylogarithms

TL;DR

This paper investigates the asymptotic behavior of Mordell-Tornheim multiple series and their integral analogues, revealing new relations among multiple polylogarithms through asymptotic analysis and integral comparisons.

Contribution

It introduces a novel approach linking multiple series and their integral analogues using Abel's summation, and uncovers new relations among multiple polylogarithms.

Findings

Established relations between multiple series and integrals using Abel's summation

Derived asymptotic formulas for the integral analogue at x=0

Discovered new relations among multiple polylogarithms

Abstract

We study the asymptotic behavior of a multiple series of Mordell-Tornheim type and its integral analogue at x=0. Our approach is to show a relation between the multiple series and its integral analogue by using Abel's summation formula, and to deeply investigate the behavior of the integral analogue. Additionally, we establish some nontrivial relations among multiple polylogarithms by comparing two seemingly different asymptotic formulas for the integral analogue.