Normal ordering of degenerate integral powers of number operator and its applications

TL;DR

This paper explores the normal ordering of degenerate integral powers of the number operator using degenerate Stirling numbers, deriving new formulas and applications in quantum operator algebra.

Contribution

It introduces a degenerate version of normal ordering for the number operator and derives related equations and formulas involving degenerate Stirling and Bell numbers.

Findings

Derived equations for degenerate Stirling numbers of the second kind.

Established a Dobinski-like formula for degenerate Bell polynomials.

Extended normal ordering techniques to degenerate cases.

Abstract

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. As a `degenerate version' of this, we consider the normal ordering of a degenerate integral power of the number operator in terms of boson operators, which is represented by means of the degenerate Stirling numbers of the second kind. As an application of this normal ordering, we derive two equations defining the degenerate Stirling numbers of the second kind and a Dobinski-like formula for the degenerate Bell polynomials.