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 ${\displaystyle \xi _{k}(s)}$ is defined by

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

where Li_k is the k-th polylogarithm

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

Properties

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

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

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

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

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

where

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

References

• 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.