 ISSAC 96 - Hypergeometric Function Representations Contents News View

 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           