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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.

Definition

The zeta function ξ k ( s ) {\displaystyle \xi _{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   {\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\ }

where Lik is the k-th polylogarithm

L i k ( z ) = ∑ n = 1 ∞ z n n k   . {\displaystyle \mathrm {Li} _{k}(z)=\sum _{n=1}^{\infty }{\frac {z^{n}}{n^{k}}}\ .}

Properties

The integral converges for ℜ ( s ) > 0 {\displaystyle \Re (s)>0} and ξ k ( s ) {\displaystyle \xi _{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 ) {\displaystyle \xi _{1}(s)=s\zeta (s+1)} where ζ {\displaystyle \zeta } is the Riemann zeta-function.

The special case s = 1 remarkably also gives ξ k ( 1 ) = ζ ( k + 1 ) {\displaystyle \xi _{k}(1)=\zeta (k+1)} where ζ {\displaystyle \zeta } 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 ) {\displaystyle \xi _{k}(m)=\zeta _{m}^{*}(k,1,\ldots ,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   . {\displaystyle \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

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