The ladder operator normal ordering problem for quantum confined systems and the generalization of the Stirling and Bell numbers

We solve the normal ordering problem for the ladder operators B-hat+ and B-hat- for strings in the form (B-hat+rB-hat-s)n for supersymmetric and shape-invariant potential systems, with r, s and n positive integers. We provide exact and explicit expressions for their normal form N{(B-hat+rB-hat-s)n}, where N{...} all B-hat- are at the right side, and find that the solution involves an expansion coefficients sequence which, for r, s > 1, corresponds to the generalization, for any shape-invariant potential system, of the classical Stirling and Bell numbers. We show that these numbers are obtained for families of polynomials generated with parameters related to the forms of the supersymmetric partner potentials. We apply the general formalism to Poeschl-Teller, Morse, Scarf and self-similar potential systems.