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

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