Some identities on degenerate r-stirling numbers via boson operators

TL;DR

This paper derives new identities and recurrence relations for degenerate r-Stirling numbers of both kinds using boson operators, linking combinatorial numbers with quantum operator algebra.

Contribution

It introduces novel identities and recurrence relations for degenerate r-Stirling numbers via boson operator techniques, expanding their algebraic and combinatorial understanding.

Findings

Derived identities and recurrence relations for degenerate r-Stirling numbers.

Established normal ordering formulas involving boson operators.

Connected degenerate r-Stirling numbers with quantum operator algebra.

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.