New Determinantal Identities on Stirling Numbers

Abstract

The Stirling numbers of first kind and Stirling numbers of second kind, denoted by \(s(n,k)\) and \(S(n,k)\) respectively, arise in a variety of combinatorial contexts. There are several algebraic and combinatorial relationships between them. Here, we state and prove four new identities concerning the determinants of matrices whose entries are unsigned Stirling numbers of first kind and Stirling numbers of second kind. We also observe an interrelationship between them based on our identities.