title: Arakawa–Kaneko zeta function
text: # 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}{\Gamma (s)}}\int _{0}^{+\infty }{\frac {t^{s-1}}{e^{t}-1}}\mathrm {Li} _{k}(1-e^{-t})\,dt\$
  
  where $Li_k$ is the k-th polylogarithm
  
      $\mathrm {Li} _{k}(z)=\sum _{n=1}^{\infty }{\frac {z^{n}}{n^{k}}}\ .$
  
  ## Properties
  
  The integral converges for $\Re (s)>0$ 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
  
      $\xi _{k}(m)=\zeta _{m}^{*}(k,1,\ldots ,1)$
  
  where
  
      $\zeta _{n}^{*}(k_{1},\dots ,k_{n-1},k_{n})=\sum _{0<m_{1}<m_{2}<\cdots <m_{n}}{\frac {1}{m_{1}^{k_{1}}\cdots m_{n-1}^{k_{n-1}}m_{n}^{k_{n}}}}\ .$
  