Arakawa–Kaneko zeta function

In mathematics, the Arakawa–Kaneko zeta function is a generalisation of the Riemann zeta function which generates special values of the polylogarithm function.

Definition

The zeta function $\xi _{k}(s)$ is defined by

\xi _{k}(s)={\frac {1}{\Gamma (s)}}\int _{0}^{+\infty }{\frac {t^{s-1}}{e^{t}-1}}\mathrm {Li} _{k}(1-e^{-t})\,dt\

where Li_k is the k-th polylogarithm

\mathrm {Li} _{k}(z)=\sum _{n=1}^{\infty }{\frac {z^{n}}{n^{k}}}\ .

The integral converges for $\Re (s)>0$ and $\xi _{k}(s)$ has analytic continuation to the whole complex plane as an entire function.

The special case k = 1 gives $\xi _{1}(s)=s\zeta (s+1)$ where $\zeta$ is the Riemann zeta-function.

The special case s = 1 remarkably also gives $\xi _{k}(1)=\zeta (k+1)$ where $\zeta$ is the Riemann zeta-function.

The values at integers are related to multiple zeta function values by

\xi _{k}(m)=\zeta _{m}^{*}(k,1,\ldots ,1)

where

\zeta _{n}^{*}(k_{1},\dots ,k_{n-1},k_{n})=\sum _{0<m_{1}<m_{2}<\cdots <m_{n}}{\frac {1}{m_{1}^{k_{1}}\cdots m_{n-1}^{k_{n-1}}m_{n}^{k_{n}}}}\ .

Kaneko, Masanobou (1997). "Poly-Bernoulli numbers". J. Théor. Nombres Bordx. 9: 221–228. Zbl 0887.11011.
Arakawa, Tsuneo; Kaneko, Masanobu (1999). "Multiple zeta values, poly-Bernoulli numbers, and related zeta functions". Nagoya Math. J. 153: 189–209. MR 1684557. Zbl 0932.11055.
Coppo, Marc-Antoine; Candelpergher, Bernard (2010). "The Arakawa–Kaneko zeta function". Ramanujan J. 22: 153–162. Zbl 1230.11106.