Explicit formulas of the relation between multiple zeta functions of Arakawa-Kaneko and Euler-Zagier types

TL;DR

This paper provides explicit formulas linking Arakawa-Kaneko and Euler-Zagier multiple zeta functions, enhancing understanding of their relationship and including functional equations for multi-polylogarithm functions.

Contribution

It offers explicit formulas for the relation between two types of multiple zeta functions and introduces functional equations for multi-polylogarithms.

Findings

Explicit formulas for the relation between Arakawa-Kaneko and Euler-Zagier zeta functions

Functional equations among multi-polylogarithm functions

Representation of Arakawa-Kaneko zeta functions as linear combinations of Euler-Zagier functions

Abstract

Multiple zeta functions of Arakawa-Kaneko and Euler-Zagier types are known as generalizations of the Riemann zeta function. In 2018, Kaneko and Tsumura proved that the multiple zeta functions of Arakawa-Kaneko type can be expressed as a ${\mathbb Q}$-linear combination of products of the ones of Euler-Zagier type and multiple zeta values. In this paper, we give explicit formulas to the above mentioned relation. Moreover, as a key of its proof, we also give certain functional equations among multi-polylogarithm functions explicitly.