Some results on Arakawa-Kaneko, Kaneko-Tsumura functions and related functions

TL;DR

This paper explores special relations and explicit forms of Arakawa-Kaneko and Kaneko-Tsumura functions, introducing a level m analogue and enhancing understanding of their properties in the context of multiple polylogarithms.

Contribution

It provides explicit formulas for these functions and introduces a new level m analogue of the Arakawa-Kaneko zeta functions, expanding the theoretical framework.

Findings

Derived explicit forms of ${ m A}(k_1, dots,k_r;z)$ and $ ext{ m extpsi}(k_1, dots,k_r;s)$

Established relations between these functions and multiple polylogarithms

Introduced a level m analogue of Arakawa-Kaneko zeta functions

Abstract

Recently, the level two analogue of multiple polylogarithm function ${\rm A}(k_1,\ldots,k_r;z)$ and Arakawa-Kaneko zeta function $\psi(k_1,\ldots,k_r;s)$ were introduced by M. Kaneko and H. Tsumura, for $k_1,\ldots,k_r \in \mathbb{Z}_{\ge 1}$ . In this paper, we investigate some of their special relations. In particular, we prove some explicit forms of ${\rm A}(k_1,\ldots,k_r;z)$ and $\psi(k_1,\ldots,k_r;s)$. Also, we introduce a level $m$ anlogue of the Arakawa-Kaneko zeta functions.