Some Identities on Degenerate \(r\)-Stirling Numbers via Boson Operators

Abstract

Broder introduced the \(r\)-Stirling numbers of the first kind and of the second kind which enumerate restricted permutations and respectively restricted partitions, the restriction being that the first \(r\) elements must be in distinct cycles and respectively in distinct subsets. Kim–Kim–Lee–Park constructed the degenerate \(r\)-Stirling numbers of both kinds as degenerate versions of them. The aim of this paper is to derive some identities and recurrence relations for the degenerate \(r\)-Stirling numbers of the first kind and of the second kind via boson operators. In particular, we obtain the normal ordering of a degenerate integral power of the number operator multiplied by an integral power of the creation boson operator in terms of boson operators where the degenerate \(r\)-Stirling numbers of the second kind appear as the coefficients.