Degenerate r-Bell polynomials arising from degenerate normal odering

TL;DR

This paper explores properties of degenerate r-Bell polynomials and numbers using boson operators, deriving new generating functions, recurrence relations, and Dobinski-like formulas through degenerate normal ordering.

Contribution

It introduces novel expressions and formulas for degenerate r-Bell polynomials and numbers based on boson operator techniques and degenerate normal ordering.

Findings

Two new generating function expressions for degenerate r-Bell polynomials

Recurrence relation for degenerate r-Bell numbers

Dobinski-like formula involving degenerate r-Stirling numbers

Abstract

Recently, Kim-Kim introduced the degenerate r-Bell polynomials and investigated some results which are derived from umbral calculus. The aim of this paper is to study some properties of the degenerate r-Bell polynomials and numbers via boson operators. In particular, we obtain two expressions for the generating function of the degenerate r-Bell polynomials in |z| , and a recurrence relation and Dobinski-like formula for the degenerate r-Bell numbers. These are derived from the degenerate normal ordering of a degenerate integral power of the number operator in terms of boson operators where the degenerate r-Stirling numbers of the second kind appear as the coefficients.