Generalized $q$-Stirling numbers and normal ordering

The normal ordering coefficients of strings consisting of $V,U$ which satisfy $UV=qVU+h{V}^s$ ($s\in \mathbb{N}$) 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.