2016/12/31 by Matthew A Williams, Williams, Matthew A
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1701.00044
openalex publication_date 2016/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.