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

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The zeta function   is defined by

 

where Lik is the k-th polylogarithm

 

Properties

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The integral converges for   and   has analytic continuation to the whole complex plane as an entire function.

The special case k = 1 gives   where   is the Riemann zeta-function.

The special case s = 1 remarkably also gives   where   is the Riemann zeta-function.

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

 

where

 

References

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  • 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.