Degenerate r-Whitney numbers and degenerate r-Dowling polynomials via Boson operators

TL;DR

This paper explores the properties and identities of degenerate r-Whitney numbers and degenerate r-Dowling polynomials, linking them to boson operator normal ordering and extending their mathematical framework.

Contribution

It introduces new properties, recurrence relations, and identities for degenerate r-Whitney numbers and degenerate r-Dowling polynomials, connecting them with boson operators.

Findings

Derived properties and recurrence relations for degenerate r-Whitney numbers.

Established orthogonality and identities related to these numbers.

Extended the framework to include degenerate r-Dowling polynomials.

Abstract

Dowling showed that the Whitney numbers of the first kind and of the second kind satisfy Stirling number-like relations. Recently, Kim-Kim introduced the degenerate r-Whitney numbers of the first kind and of the second kind, as degenerate versions and further generalizations of the Whitney numbers of both kinds. The normal ordering of an integral power of the number operator in terms of boson operators is expressed with the help of the Stirling numbers of the second kind. In this paper, it is noted that the normal ordering of a certain quantity involving the number operator is expressed in terms of the degenerate r-Whitney numbers of the second kind. We derive some properties, recurrence relations, orthogonality relations and several identities on those numbers from such normal ordering. In addition, we consider the degenerate r-Dowling polynomials as a natural extension of the degenerate r-Whitney numbers of the second kind and investigate their properties.