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

We study the (p, q)-deformed generalized Weyl algebra generated by variables X, Y and Zₚ satisfying the (p, q)-commutation relations XY-qYX=h YˢZₚ, XZₚ=pZₚX , and ZₚY=pYZₚ , with s∈ N₀ . 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.