On functions of Arakawa and Kaneko and multiple zeta functions

TL;DR

This paper investigates functions related to multiple zeta functions introduced by Arakawa and Kaneko, establishing their properties and connections, and providing an alternative proof of a known result by Ohno.

Contribution

It offers new insights into the functions of Arakawa and Kaneko, relating them to multiple zeta functions and presenting an alternative proof of Ohno's result.

Findings

Established connections between Arakawa-Kaneko functions and multiple zeta functions

Provided an alternative proof of Ohno's theorem

Enhanced understanding of the properties of these special functions

Abstract

We study for $s\in\N=\{1,2,...\}$ the functions $\xi_{k}(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}}{e^t-1}\Li_{k}(1-e^{-t})dt$, and more generally $\xi_{k_1,...,k_r}(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}}{e^t-1}\Li_{k_1,...,k_r}(1-e^{-t})dt$, introduced by Arakawa and Kaneko \cite{Arakawa} and relate them with (finite) multiple zeta functions, partially answering a question of \cite{Arakawa}. In particular, we give an alternative proof of a result of Ohno \cite{Ohno2}.