Generalized Arakawa-Kaneko zeta functions

TL;DR

This paper introduces and studies a generalized Arakawa-Kaneko zeta function, exploring its properties, special values, and potential applications, extending the classical zeta functions with new parameters and functional forms.

Contribution

It defines a new class of zeta functions involving multiple polylogarithms and investigates their properties and applications, expanding the theory of special functions.

Findings

Derived functional equations and analytic properties.

Identified special values and relations to known zeta functions.

Presented potential applications in mathematical analysis.

Abstract

Let $p,x$ be real numbers, and $s$ be a complex number, with $\Re(s)>1-r$, $p\geq 1$, and $x+1>0$. The zeta function $Z^{\bf\alpha}_p(s;x)$ is defined by $$ Z^{\bf\alpha}_p(s;x) =\frac{1}{\Gamma(s)}\int^\infty_0 \frac{e^{-xt}} {e^t-1}\,Li_{\bf{\alpha}}\left(\frac{1-e^{-t}}p\right) t^{s-1}\,dt, $$ where ${\bf\alpha}=(\alpha_1,\ldots,\alpha_r)$ is a $r$-tuple positive integers, and $Li_{\bf{\alpha}}(z)$ is the one-variable multiple polylogarithms. Since $Z^{\bf\alpha}_1(s;0)=\xi(\bf\alpha;s)$, we call this function as a generalized Arakawa-Kaneko zeta function. In this paper, we investigate the properties and values of $Z^{\bf\alpha}_p(s;x)$ with different values $s$, $x$, and $p$. We then give some applications on them.