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

: $\zeta_n^*(k_1,\dots,k_{n-1},k_n)=\sum_{0$

{{cite journal | last1=Kaneko | first1=Masanobou | title=Poly-Bernoulli numbers | zbl=0887.11011 | journal= Journal de Théorie des Nombres de Bordeaux| volume=9 | pages=221–228 | year=1997 | doi=10.5802/jtnb.197 | hdl=2324/21658 | hdl-access=free }}
{{cite journal | last1=Arakawa | first1=Tsuneo | last2=Kaneko | first2=Masanobu | title=Multiple zeta values, poly-Bernoulli numbers, and related zeta functions | journal= Nagoya Mathematical Journal| volume=153 | pages=189–209 | year=1999 | doi=10.1017/S0027763000006954 | zbl=0932.11055 | mr=1684557 | url=http://projecteuclid.org/euclid.nmj/1114630825 | hdl=2324/20424 | hdl-access=free }}
{{cite journal | last1=Coppo | first1=Marc-Antoine | last2=Candelpergher | first2=Bernard | title=The Arakawa–Kaneko zeta function | zbl=1230.11106 | journal= The Ramanujan Journal| volume=22 | pages=153–162 | year=2010 | issue=2 | doi=10.1007/s11139-009-9205-x }}