2022/04/16 by Taekyun Kim, Kim, Taekyun, Dae San Kim +1
Mathematics · #11B73 #11B83 #Advanced Mathematical Identities #FOS: Mathematics #Fractional Differential Equations Solutions #Mathematical Inequalities and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2204.07843
openalex publication_date 2022/04/16 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
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.