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 Lik 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


 * $$\zeta_n^*(k_1,\dots,k_{n-1},k_n)=\sum_{0<m_1<m_2<\cdots<m_n}\frac{1}{m_1^{k_1}\cdots m_{n-1}^{k_{n-1}}m_n^{k_n}} \ . $$