Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. I: Algebraic Framework and Combinatorial Identities

The $(p,q)$-deformed generalized Weyl algebra is generated by variables $X,Y$ and $Z_p$ which satisfy the commutation relations $XY-qYX=h{Y}^s{Z}_p,X{Z}_p=p{Z}_{p}X$, and $Z_{p}Y=pY{Z}_p$, with $s\in {\mathbb{N}}_0$. We investigate the problem of normal ordering arbitrary words in these letters with the help of Young diagrams, and we treat certain special cases explicitly. In particular, the connection to generalized Stirling numbers is considered in detail.