Generalized $q$-Stirling numbers and normal ordering

TL;DR

This paper explores generalized $q$-Stirling numbers and their connection to normal ordering, rook numbers, and Bell numbers, providing new formulas, recurrences, and extensions for these combinatorial quantities.

Contribution

It introduces a unified framework for generalized $q$-Stirling numbers, linking them to rook numbers and Bell numbers, with explicit formulas and extensions to real parameters.

Findings

Derived recurrences and explicit formulas for generalized $q$-Stirling numbers.

Established connections between these numbers and rook numbers under a new rule.

Extended rook number models to real parameters s.

Abstract

The normal ordering coefficients of strings consisting of $V,U$ which satisfy $UV=qVU+hV^s$ ($s\in\mathbb N$) are considered. These coefficients are studied in two contexts: first, as a multiple of a sequence satisfying a generalized recurrence, and second, as $q$-analogues of rook numbers under the row creation rule introduced by Goldman and Haglund. A number of properties are derived, including recurrences, expressions involving other $q$-analogues and explicit formulas. We also give a Dobinsky-type formula for the associated Bell numbers and the corresponding extension of Spivey's Bell number formula. The coefficients, viewed as rook numbers, are extended to the case $s\in\mathbb R$ via a modified rook model.