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

作者：Aleixo A N F^*; Balantekin A B

来源：Journal of Mathematical Physics, 2015, 56(12): 122108.

DOI：10.1063/1.4936954

We resolve the normal ordering problem for symmetric ((D) over cap (+)(D) over cap (-))(n) and asymmetric ((D) over cap (r)(+)(D) over cap (-))(n) strings of the nonlinear deformed ladder operators (D) over cap (+/-) for super-symmetric and shape-invariant potential systems, where r and n are positive integers. We provide exact and explicit expressions for their normal forms N{((D) over cap (+)(D) over cap (-))(n)} and N{((D) over cap (r)(+)(D) over cap (-))(n)}, where in N{...} all (D) over cap (-)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.

• 出版日期2015-12