Factorization of the hypergeometric-type difference equation on non-uniform lattices: dynamical algebra

Abstract

We argue that one can factorize the difference equation of hypergeometric type on the nonuniform lattices in general case. It is shown that in the most cases of q-linear spectrum of the eigenvalues this directly leads to the dynamical symmetry algebra su q (1, 1), whose generators are explicitly constructed in terms of the difference operators, obtained in the process of factorization. Thus all models with the q-linear spectrum (some of them, but not all, previously considered in a number of publications) can be treated in a unified form.

Factorization Method and General Second Order Linear Difference Equation

“…It is well known that the above equation (43) describes classical orthogonal polynomials of a discrete variable such as the Charlier, Meixner, Kravchuk, Hahn polynomials. See [24,2] for more details.…”

“…This work is an extension of a previous work [10]. Some results obtained in [1,8,9,16] are used, and adapted to our context. The paper is organized as follows.…”

Recurrence relations for discrete hypergeometric functions

“…[3,6,7,15,21] and reference therein) that are related with the q-polynomials. For these cases only a few recurrences are known [4].…”

“…Our main aim in this paper is to present a constructive approach for generating recurrence relations and ladder-type operators for some discrete system such as the discrete oscillators [2,5,6,8,9,10,11,12,13,14], discrete Calogero-Sutherland model [20], etc. The main idea is to use the connection of the wave functions with the classical discrete polynomials in a similar way as it was done in our previous paper [16] for the N -th dimensional oscillators and hydrogenlike atoms.…”

On a q-extension of the linear harmonic oscillator with the continuous orthogonality property on ℝ

“…But we could have started equivalently with the q-difference equation (3.5) itself and have directly factorized it in terms of the same operators a(x; q) and a † (x; q) (for a more detailed discussion of the factorization of difference equations, see, for example, [15,16]). So we have established that our q-model is governed by the Hamiltonian (3.6), which admits the factorization (3.11) in terms of the operators a(x; q) and a † (x; q), satisfying the qcommutation relation (3.12).…”