Normal ordering for nonlinear deformed ladder operators and the f-generalization of the Stirling and Bell numbers

Abstract

We resolve the normal ordering problem for symmetric (Dˆ+Dˆ−)n and asymmetric (Dˆ+rDˆ−)n strings of the nonlinear deformed ladder operators Dˆ± for supersymmetric and shape-invariant potential systems, where r and n are positive integers. We provide exact and explicit expressions for their normal forms N{(Dˆ+Dˆ−)n} and N{(Dˆ+rDˆ−)n}, where in N{...} all Dˆ− are at the right side. We find that the solutions involve sequence of expansion coefficients which, for r, n > 1, corresponds to the f-deformed generalization of the classical Stirling and Bell numbers of the second kind. We apply the general formalism for the translational shape-invariant potential systems as well as for the particular case of the harmonic oscillator potential system. We show that these numbers are obtained for families of polynomial expressions generated with the deformations parameters and the parameters related to the forms of the supersymmetric partner potentials.