Difference between revisions of "Lerch transcendent"
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Revision as of 00:07, 2 April 2015
The Lerch transcendent $\Phi$ is defined by $$\Phi(z,s,a)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{(a+k)^s}.$$
Properties
Theorem
The following formula holds: $$\Phi(z,n,1)=\dfrac{\mathrm{Li}_n(z)}{z},$$ where $\Phi$ denotes the Lerch transcendent and $\mathrm{Li_n}$ denotes the polylogarithm.
Proof
References
Theorem
The following formula holds: $$\Phi(e^{2\pi i \lambda},z,a)=L(\lambda,a,z),$$ where $\Phi$ denotes the Lerch transcendent and $L$ denotes the Lerch zeta function.
