Arakawa-Kaneko zeta function

Arakawa-Kaneko zeta functions are defined by $$\xi_k(s)=\dfrac{1}{\Gamma(s)} \displaystyle\int_0^{\infty} \dfrac{t^s-1}{e^t-1} \mathrm{Li}_k(1-e^{-t})dt,$$ where $\Gamma$ denotes the gamma function and $\mathrm{Li}_k$ denotes the polylogarithm.

Properties

Theorem: The integral defining $\xi_k$ converges for $\mathrm{Re}(s)>0$ and $\xi_k$ has analytic continuation to $\mathbb{C}$ as an entire function.

Proof: █

Propositon: If $k=1$, then the following formula holds: $$\xi_1(s)=s\zeta(s+1),$$ where $\xi_1$ denotes the Arakawa-Kaneko zeta function and $\zeta$ denotes the Riemann zeta function.

Proof: █