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 [math]\displaystyle{ \xi_k(s) }[/math] is defined by

[math]\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 \ }[/math]

where Li_k is the k-th polylogarithm

[math]\displaystyle{ \mathrm{Li}_k(z) = \sum_{n=1}^{\infty} \frac{z^n}{n^k} \ . }[/math]

Properties

The integral converges for [math]\displaystyle{ \Re(s) \gt 0 }[/math] and [math]\displaystyle{ \xi_k(s) }[/math] has analytic continuation to the whole complex plane as an entire function.

The special case k = 1 gives [math]\displaystyle{ \xi_1(s) = s \zeta(s+1) }[/math] where [math]\displaystyle{ \zeta }[/math] is the Riemann zeta-function.

The special case s = 1 remarkably also gives [math]\displaystyle{ \xi_k(1) = \zeta(k+1) }[/math] where [math]\displaystyle{ \zeta }[/math] is the Riemann zeta-function.

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

[math]\displaystyle{ \xi_k(m) = \zeta_m^*(k,1,\ldots,1) }[/math]

where

[math]\displaystyle{ \zeta_n^*(k_1,\dots,k_{n-1},k_n)=\sum_{0\lt m_1\lt m_2\lt \cdots\lt m_n}\frac{1}{m_1^{k_1}\cdots m_{n-1}^{k_{n-1}}m_n^{k_n}} \ . }[/math]

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.