Difference between revisions of "Arakawa-Kaneko zeta function"
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Arakawa-Kaneko zeta functions are defined by | Arakawa-Kaneko zeta functions are defined by | ||
| − | $$\xi_k(s)=\dfrac{1}{\Gamma(s)} \displaystyle\int_0^{\infty} \dfrac{t^s-1}{e^t-1} \mathrm{Li}_k(1-e^{-t}) | + | $$\xi_k(s)=\dfrac{1}{\Gamma(s)} \displaystyle\int_0^{\infty} \dfrac{t^s-1}{e^t-1} \mathrm{Li}_k(1-e^{-t}) \mathrm{d} t,$$ |
where $\Gamma$ denotes the [[gamma function]] and $\mathrm{Li}_k$ denotes the [[polylogarithm]]. | where $\Gamma$ denotes the [[gamma function]] and $\mathrm{Li}_k$ denotes the [[polylogarithm]]. | ||
Latest revision as of 17:22, 24 June 2016
Arakawa-Kaneko zeta functions are defined by $$\xi_k(s)=\dfrac{1}{\Gamma(s)} \displaystyle\int_0^{\infty} \dfrac{t^s-1}{e^t-1} \mathrm{Li}_k(1-e^{-t}) \mathrm{d} t,$$ where $\Gamma$ denotes the gamma function and $\mathrm{Li}_k$ denotes the polylogarithm.
Properties
Theorem: The integral defining $\xi_k$ converges for $\mathrm{Re}(s)>0$ and $\xi_k$ has analytic continuation to $\mathbb{C}$ as an entire function.
Proof: █
Propositon: If $k=1$, then the following formula holds: $$\xi_1(s)=s\zeta(s+1),$$ where $\xi_1$ denotes the Arakawa-Kaneko zeta function and $\zeta$ denotes the Riemann zeta function.
Proof: █
