Recurrence relation

Gene

Quantum mechanics

Stirling number

Pure mathematics

Discrete mathematics

Degenerate energy levels

Biochemistry

Physics

Mathematics

Natural number

Stirling numbers of the first kind

Transcription factor

Combinatorics

Repressor

Chemistry

Operator (biology)

Generating function

Algebra over a field

Boson

Advances in Applied Mathematics

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.

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