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 $Z_p$ satisfying the $(p,q)$-commutation relations $XY-qYX=h{Y}^s{Z}_p,X{Z}_p=p{Z}_{p}X$, and $Z_{p}Y=pY{Z}_p$, 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.