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
