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

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