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

Abstract

We study the (p, q)-deformed generalized Weyl algebra generated by variables X, Y and Zp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_p$$\end{document} satisfying the (p, q)-commutation relations XY-qYX=hYsZp,XZp=pZpX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$$\end{document}, and ZpY=pYZp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z_pY=pYZ_p$$\end{document}, with s∈N0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in \mathbb {N}_0$$\end{document}. Within this framework, we investigate the noncommutative binomial formula (X+Y)n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X+Y)^n$$\end{document} 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.