Ladder operators' normal ordering problem for quantum-deformed systems and the (q, p)-generalization of the Stirling and Bell numbers

Abstract

We resolve the ladder operators normal ordering problem for strings in the form for (q, p)-deformed supersymmetric and shape-invariant potential systems, where n is a positive integer. We provide exact and explicit expressions for their normal form , where in all are at the right-hand side, and find that the solution involves expansion-coefficients sequence which corresponds to the generalization, for any (q, p)-deformed shape-invariant potential system, of the classical Stirling and Bell numbers. We show that these numbers are given for expressions depending on parameters related to the forms of supersymmetric partner potentials and the system quantum deformation. We apply the general formalism to Pöschl–Teller, Morse and Scarf potential systems.