ISSAC 96 - Hypergeometric Function Representations
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 Lerch Phi and Polylogarithms

If the coefficients of the series representation of a hypergeometric function are rational functions of the summation index, then the hypergeometric function can be expressed as a linear sum of Lerch functions.  The Lerch function is defined by

Further, if the parameters of the hypergeometric function are rational, we can proceed to express the hypergeometric function as a linear sum of polylogarithms.  The polylogarithm function is defined by

The first theorem shows how to express such a hypergeometric function as a linear sum of Lerch functions.

Theorem Let , , and

have partial fraction decomposition

Then

The next theorem can be used to range reduce the third argument of a Lerch into the interval

Theorem

The next two theorems show how to convert Lerch into polylogarithms if the third argument is rational.

Theorem

Theorem Let and .  Then

Corollary Let and .  Then

where

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