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

Recently, the level two analogue of multiple polylogarithm function ${\rm A}(k_1,\ldots,k_r;z)$andArakawa-Kanekozetafunction$\psi(k_1,\ldots,k_r;s)$wereintroducedbyM\mathrm{.}KanekoandH\mathrm{.}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$\psi(k_1,\ldots,k_r;s)$\mathrm{.}Also,weintroducealevel$m$ anlogue of the Arakawa-Kaneko zeta functions.