Some Identities on Degenerate r-Stirling Numbers via Boson Operators

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. DOI 10.1134/S1061920822040094