On Two Applications of Herschel's Theorem

TL;DR

This paper explores two applications of Herschel's theorem: providing elementary proofs of combinatorial numbers and polynomials, and extending the polylogarithmic function analytically beyond its usual convergence circle.

Contribution

It offers new elementary proofs for known combinatorial expressions and advances the understanding of polylogarithms through analytical continuation.

Findings

Elementary proofs for combinatorial numbers and polynomials.

Extended the domain of polylogarithmic functions beyond convergence circle.

Enhanced understanding of Herschel's theorem applications.

Abstract

As a first application of a very old theorem, known as Herschel's theorem, we provide direct elementary proofs of several explicit expressions for some numbers and polynomials that are known in combinatorics. The second application deals with the analytical continuation of the polylogarithmic function of complex argument beyond the circle of convergence.