On values of zeta functions of Arakawa-Kaneko type

TL;DR

This paper generalizes Arakawa-Kaneko type zeta functions to include mixed positive and negative integer indices and expresses their positive integer values using multiple Hurwitz zeta star values.

Contribution

It introduces a new class of zeta functions with mixed indices and establishes their values at positive integers in terms of multiple Hurwitz zeta star values.

Findings

Values at positive integers are expressed via multiple Hurwitz zeta star values.

Generalization includes both positive and negative indices.

Extends previous work on Arakawa-Kaneko zeta functions.

Abstract

For these two decades, the Arakawa-Kaneko zeta function has been studied actively. Recently Kaneko and Tsumura constructed its variants from the viewpoint of poly-Bernoulli numbers. In this paper, we generalize their zeta functions of Arakawa-Kaneko type to those with indices in which positive and negative integers are mixed. We show that values of these functions at positive integers can be expressed in terms of the multiple Hurwitz zeta star values.