Lerch transcendent


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where the terms with i + a = 0 are excluded. 

h19letr1.gif (1824 bytes)The Lerch transcendent Φ(x1, x2, a) is a generalization of the polylogarithm (regarding x1) and the (generalized) zeta function (regarding x2).

Many infinite sums (of reciprocal powers) can expressed as a Lerch transcendent, for instance the Catalan beta function.
The function is also related to integrals of the Fermi-Dirac distribution. And it is used to evaluate some series in number theory (Dirichlet L-series).