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

Abstract

Recently, the level two analogue of the multiple polylogarithm function \(\textrm{A}(k_1,\ldots ,k_r;z)\) and the Arakawa–Kaneko zeta function \(\psi (k_1,\ldots ,k_r;s)\) have been 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 \(\textrm{A}(k_1,\ldots ,k_r;z)\) and \(\psi (k_1,\ldots ,k_r;s)\). Also, we introduce a level m analogue of the Arakawa–Kaneko zeta functions.