Normal Ordering for Deformed Boson Operators and Operator-valued Deformed Stirling Numbers

TL;DR

This paper extends normal ordering formulas to deformed bosons, introducing operator-valued deformed Stirling numbers that differ between two types of deformed bosons, revealing new algebraic structures.

Contribution

It introduces and analyzes operator-valued deformed Stirling numbers for two classes of deformed bosons, expanding the mathematical framework of bosonic operator algebra.

Findings

Deformed Stirling numbers replace classical ones in normal ordering for M-type bosons.

For P-type bosons, deformed Stirling numbers depend on the number operator.

Distinct algebraic structures are identified for different deformed boson types.

Abstract

The normal ordering formulae for powers of the boson number operator $\hat{n}$ are extended to deformed bosons. It is found that for the `M-type' deformed bosons, which satisfy $a a^{\dagger} - q a^{\dagger} a = 1$, the extension involves a set of deformed Stirling numbers which replace the Stirling numbers occurring in the conventional case. On the other hand, the deformed Stirling numbers which have to be introduced in the case of the `P-type' deformed bosons, which satisfy $a a^{\dagger} - q a^{\dagger} a = q^{-\hat{n}}$, are found to depend on the operator $\hat{n}$. This distinction between the two types of deformed bosons is in harmony with earlier observations made in the context of a study of the extended Campbell-Baker-Hausdorff formula.