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

${\xi }_k(s)=\frac{1}{\mathrm{\Gamma }(s)}\underset{0}{\overset{+\mathrm{\infty }}{\int }}\frac{t^{s-1}}{e^t-1}{\mathrm{L}\mathrm{i}}_k(1-e^{-t})dt$

where Li_k is the k-th polylogarithm

${\mathrm{L}\mathrm{i}}_k(z)=\sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{n^k}.$

Properties

The integral converges for $\mathrm{\Re }(s)>0$ 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

${\xi }_k(m)={\zeta }_m^{*}(k,1,\dots ,1)$

where

${\zeta }_n^{*}(k_1,\dots ,k_{n-1},k_n)=\sum _{0<m_1<m_2<\cdots <m_n}\frac{1}{m_1^{k_1}\cdots m_{n-1}^{k_{n-1}}{m}_n^{k_n}}.$

References

• Kaneko, Masanobou (1997). "Poly-Bernoulli numbers". J. Théor. Nombres Bordx. 9: 221–228. 
• Arakawa, Tsuneo; Kaneko, Masanobu (1999). "Multiple zeta values, poly-Bernoulli numbers, and related zeta functions". Nagoya Math. J. 153: 189–209. http://projecteuclid.org/euclid.nmj/1114630825. 
• Coppo, Marc-Antoine; Candelpergher, Bernard (2010). "The Arakawa–Kaneko zeta function". Ramanujan J. 22: 153–162.