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
The next theorem can be used to range reduce the third argument of a Lerch into the interval .
The next two theorems show how to convert Lerch into polylogarithms if the third argument is rational.
Theorem Let and . Then
Corollary Let and . Then
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