Stirling Functions and a Generalization of Wilson's Theorem

TL;DR

This paper introduces Stirling functions extending the second kind Stirling numbers, explores their properties, and generalizes Wilson's Theorem by analyzing divisibility and roots of associated polynomials.

Contribution

It defines Stirling functions S(m,n,z), characterizes their roots, and extends Wilson's Theorem through polynomial divisibility analysis.

Findings

Solutions for real z are explicitly computed.

All real roots of polynomial P(m,n,z) are simple.

The generalization of Wilson's Theorem is established.

Abstract

For positive integers m and n, denote S(m,n) as the associated Stirling number of the second kind and let z be a complex variable. In this paper, we introduce the Stirling functions S(m,n,z) which satisfy S(m,n,z) = S(m,n) for any z which lies in the zero set of a certain polynomial P(m,n,z). For all real z, the solutions of S(m,n,z) = S(m,n) are computed and all real roots of the polynomial P(m,n,z) are shown to be simple. Applying the properties of the Stirling functions, we investigate the divisibility of the numbers S(m,n) and then generalize Wilson's Theorem.