Stirling Numbers and Inverse Factorial Series

Abstract

We discuss inverse factorial series and their relation to Stirling numbers of the first kind. We prove a special representation of the polylogarithm function in terms of series with such Stirling numbers. Using various identities for Stirling numbers of the first kind we construct a number of expansions of functions in terms of inverse factorial series where the coefficients are special numbers. These results are used to reprove the asymptotic expansion of some classical functions. We also prove a binomial formula involving inverse factorials.