Generalized Arakawa-Kaneko zeta functions

Abstract

Let $p,x$ be real numbers, and $s$ be a complex number, with $\Re (s)>1-r$, $p\ge 1$, and $x+1>0$. The zeta function $Z_p^{\alpha }(s;x)$ is defined by $$Z_p^{\alpha }(s;x)=\frac{1}{\Gamma (s)}{\int }_0^{\infty }\frac{e^{-xt}}{e^t-1}L{i}_{\alpha }\left(\frac{1-e^{-t}}{p}\right)t^{s-1}dt,$$ where $\alpha =({\alpha }_1,\dots ,{\alpha }_r)$ is a $r$-tuple positive integers, and $L{i}_{\alpha }(z)$ is the one-variable multiple polylogarithms. Since $Z_1^{\alpha }(s;0)=\xi (\alpha ;\mathbf{s})$, we call this function as a generalized Arakawa-Kaneko zeta function. In this paper, we investigate the properties and values of $Z_p^{\alpha }(s;x)$ with different values $s$, $x$, and $p$. We then give some applications on them.

20 pages

References in corpus (2)

• On functions of Arakawa and Kaneko and multiple zeta functions
• Multinomial Sum Formulas of Multiple Zeta Values

Cited by in corpus (2)

• Apéry-Type Series and Colored Multiple Zeta Values
• Explicit Evaluation of Euler-Apéry Type Multiple Zeta Star Values and Multiple $t$-Star Values