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arXiv · 2607.11693

Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. III: The binomial formula

Toufik Mansour·Lahcen Oussi·Matthias Schork
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Abstract

We study the $(p, q)$-deformed generalized Weyl algebra generated by variables $X, Y$ and $Z_p$ satisfying the $(p, q)$-commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, with $s\in \mathbb{N}_0$. Within this framework, we investigate the noncommutative binomial formula $(X+Y)^n$ and related identities. In particular, we show how the associated normal ordering coefficients can be expressed in terms of $(p,q)$-deformed $s$-rook numbers. We treat several special cases explicitly, recovering known results from literature as well as deriving new ones.

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Toufik Mansour, Lahcen Oussi, Matthias Schork. 2026-07-13. Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. III: The binomial formula. https://arxiv.org/abs/2607.11693

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