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Arakawa–Kaneko zeta function

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In mathematics, the Arakawa–Kaneko zeta function is a generalisation of the Riemann zeta function which generates special values of the polylogarithm function.

Contents

Definition

The zeta function ξ k ( s ) is defined by

ξ k ( s ) = 1 Γ ( s ) ∫ 0 + ∞ t s − 1 e t − 1 L i k ( 1 − e − t ) d t  

where Lik is the k-th polylogarithm

L i k ( z ) = ∑ n = 1 ∞ z n n k   .

Properties

The integral converges for ℜ ( s ) > 0 and ξ k ( s ) has analytic continuation to the whole complex plane as an entire function.

The special case k = 1 gives ξ 1 ( s ) = s ζ ( s + 1 ) where ζ is the Riemann zeta-function.

The special case s = 1 remarkably also gives ξ k ( 1 ) = ζ ( k + 1 ) where ζ is the Riemann zeta-function.

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

ξ k ( m ) = ζ m ∗ ( k , 1 , … , 1 )

where

ζ n ∗ ( k 1 , … , k n − 1 , k n ) = ∑ 0 < m 1 < m 2 < ⋯ < m n 1 m 1 k 1 ⋯ m n − 1 k n − 1 m n k n   .

References

Arakawa–Kaneko zeta function Wikipedia
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