Rook theory, normal ordering in the $q$-deformed Ore algebra and the polynomial generalization

TL;DR

This paper introduces a combinatorial interpretation of normal ordering coefficients in the $q$-deformed Ore algebra using rook placements, and extends these concepts to polynomial Weyl algebras.

Contribution

It provides a novel combinatorial framework for $q$-deformed Ore and polynomial Weyl algebras, including new $q$-deformed Stirling and Lah numbers and their properties.

Findings

Normal ordering coefficients relate to rook and file placements on specific boards.

Recurrence relations for $q$-deformed Ore-Stirling and Ore-Lah numbers are derived.

Normal ordered forms of binomials are explicitly determined in the $q$-deformed algebras.

Abstract

For words in the variables $X$ and $Y$ satisfying the commutation relation of the $q$-deformed generalized Ore algebra, $XY-qYX= \mu I + \nu Y$, we show that the corresponding normal ordering coefficients can be given an interpretation in terms of mixed placements of rooks and files. In particular, the associated $q$-deformed Ore-Stirling and Ore-Lah numbers are treated in detail. We show that the $q$-deformed Ore-Stirling numbers (resp., $q$-deformed Ore-Lah numbers) are given as mixed placement numbers of rooks and files on the staircase board (resp., Laguerre board). Using this combinatorial interpretation, their recurrence relations are derived. In addition, the normal ordered form of the binomial $(X+Y)^m$ in the $q$-deformed generalized Ore algebra is determined. These considerations are then extended to the $q$-deformed polynomial Weyl algebra generated by $X$ and $Y$ satisfying $XY-qYX=f(Y)$ for some polynomial $f\in \mathbb{C}[Y]$. In particular, associated $q$-deformed polynomial Stirling and Lah numbers are introduced and their properties studied. The normal ordered form of the binomial is also extended to the $q$-deformed polynomial Weyl algebra.