Mathematical Research Letters
Volume 16 (2009)
Resurgence of the fractional polylogarithms
Pages: 817 – 826
The fractional polylogarithms, depending on a complex parameter α, 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 α, 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 [CG3]. The motivic or physical interpretation of the monodromy of the fractional polylogarithms for non-integer values of α is unknown to the authors.
fractional polylogarithms, resurgence, Écalle, Mittag-Leffler decomposition, monodromy, motives, asymptotics, Appell’s equation, Lambert function
2010 Mathematics Subject Classification
Primary 57N10. Secondary 57M25.