On generalization of duality formulas for the Arakawa-Kaneko type zeta functions

Kaneko and Tsumura introduced the Arakawa-Kaneko type zeta function $\eta (-k_1,\dots ,-k_r;s_1,\dots ,s_r)$ for non-negative integers $k_1,\dots ,k_r$ and complex variables $s_1,\dots ,s_r$. Recently, Yamamoto showed that, by using the multiple integral expression, $\eta (u_1,\dots ,u_r;s_1,\dots ,s_r)$ can be extended to an analytic function of 2$r$ variables. Also, he showed that the function $\eta (u_1,\dots ,u_r;s_1,\dots ,s_r)$ satisfies a duality formula. In this paper, by using the a generalization of non-strict multi-indexed polylogarithm, we define a kind of Arakawa-Kaneko type zeta function, and show that this function satisfies a certain duality formula.