Stirling Functions and a Generalization of Wilson's Theorem

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.