Here is a representation of the hypergeometric function
in terms of the polylogarithm: For
and
, the hypergeometric series representation is given by
where
is the unsigned Stirling number of the first kind. In (a), we used the expansion of the rising Pochhammer factorial and in (b), we used the definition of the polylogarithm. Furthermore, the
follows from the analytic continuation
of the polylogarithm.