  The fractional polylogarithms, depending on a complex parameter $\a$, are
defined by a series which is analytic inside the unit disk. After an elementary
conversion of the series into an integral presentation, we show that the
fractional polylogarithms are multivalued analytic functions in the complex
plane minus 0 and 1. For non-integer values of $\a$, we prove the analytic
continuation, compute the monodromy around 0 and 1, give a Mittag-Leffler
decomposition and compute the asymptotic behavior for large values of the
complex variable. The fractional polylogarithms are building blocks of
resurgent functions that are used in proving that certain power series
associated with knotted objects are resurgent. This is explained in a separate
publication \cite{CG3}. The motivic or physical interpretation of the monodromy
of the fractional polylogarithms for non-integer values of $\a$ is unknown to
the authors.
