Stirling numbers and inverse factorial series

TL;DR

This paper explores the relationship between Stirling numbers of the first kind and inverse factorial series, providing new representations of functions and asymptotic expansions, including a binomial formula involving inverse factorials.

Contribution

It introduces novel expansions of functions using inverse factorial series with Stirling numbers and derives new identities and asymptotic formulas.

Findings

Representation of the polylogarithm function via inverse factorial series

New identities involving Stirling numbers of the first kind

Asymptotic expansions of classical functions

Abstract

We study 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 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 prove/reprove the asymptotic expansion of some classical functions. We also prove a binomial formula involving inverse factorials.