On some arithmetic properties of polynomial expressions involving Stirling numbers of the second kind

Abstract

Let S(n, k) be the classical Stirling numbers of the second kind, d > 1 be an integer, and P,Q,R ∈ Q[X1, . . . , Xm] be nonconstant polynomials such that P does not divide Q and R is not a dth power. We prove that if k1, . . . , km are any sufficiently large distinct positive integers then, setting Si = S(n, ki), Q(S1,...,Sm) P (S1,...,Sm) ∈ Z for only finitely many n ∈ N and R(S1, . . . , Sm) = xd for only finitely many pairs (n, x) ∈ N2. We extend the latter finiteness result to all triples (n, x, d) ∈ N3, x, d > 1. Our proofs are based on the results of Corvaja and Zannier. We give similar but more particular results on the more general Stirling-like numbers T (n, k).