Congruences for Weighted and Degenerate Stirling Numbers

Abstract

Let s(n,k) and S(n,k) be the (unsigned) Stirling numbers of the first and second kinds respectively [8, pp. 204–219]. In [1] and [10] congruence formulas (mod p), p prime, are proved for s(n,k) and S(n,k). As applications, the residues (mod p) for p = 2, 3, and 5 are worked out for both kinds of Stirling numbers. In [10] similar formulas are proved for the associated Stirling numbers d(n,k) and b(n,k).