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

TL;DR

This paper extends duality formulas for Arakawa-Kaneko type zeta functions using a generalized polylogarithm, broadening their analytic and duality properties.

Contribution

It introduces a generalized Arakawa-Kaneko zeta function based on multi-indexed polylogarithms and proves a new duality formula for it.

Findings

Established a new duality formula for the generalized zeta function.

Extended the analytic domain of the zeta function using multi-variable integral representations.

Connected the generalized zeta function to non-strict multi-indexed polylogarithms.

Abstract

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, Yamamoto showed that, by using the multiple integral expression, $\eta(u_1,\ldots,u_r;s_1,\ldots,s_r)$ can be extended to an analytic function of 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 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.