Some identities involving degenerate Stirling numbers arising from normal ordering

TL;DR

This paper introduces identities and recurrence relations for degenerate Stirling numbers of both kinds, derived from normal ordering in quantum boson operators, extending classical Stirling number properties to a degenerate setting.

Contribution

It presents new identities and recurrence relations for degenerate Stirling numbers, linking them to normal ordering of degenerate powers of the number operator in quantum mechanics.

Findings

Derived identities for degenerate Stirling numbers of the first and second kind.

Established recurrence relations connecting degenerate Stirling numbers to boson operator normal ordering.

Showed that degenerate Stirling numbers generalize classical Stirling numbers in quantum operator contexts.

Abstract

In this paper, we derive some identities and recurrence relations for the degenerate Stirling numbers of the first kind and of the second kind which are degenerate versions of the ordinary Stirling numbers of the first kind and of the second kind. They are deduced from the normal oderings of degenerate integral powers of the number operator and their inversions, certain relations of boson operators and from the recurrence relations of the Stirling numbers themselves. Here we note that, while the normal ordering of an integral power of the number operator is expressed with the help of the Stirling numbers of the second kind, that of a degenerate integral power of the number operator is represented by means of the degenerate Stirling numbers of the second kind.