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

Abstract

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