Factorization method for difference equations of hypergeometric type on nonuniform lattices

Abstract

We study the factorization of the hypergeometric-type difference equation of Nikiforov and Uvarov on nonuniform lattices. An explicit form of the raising and lowering operators is derived and some relevant examples are given.

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

“…Consequently, H q (s)Φ n (s) = q A straightforward calculation shows that the operators a ↑ (s) and a ↓ (s) are mutually adjoint. A similar factorization was obtained earlier in [5] and more recently in [3] with the aid of a different technique. The special case of the Al-Salam & Carlitz I polynomials are the discrete q-Hermite I h n (x; q), x = q s , polynomials, which correspond to the parameter a = −1 [20].…”

“…Consequently, the operators K ± and K 0 are such that [K 0 (s), K ± (s)] = ±K ± (s) and [K + (s), K − (s)] = 2K 0 (s). (5.11) This case corresponds to the Lie algebra so (3).…”

“…In the case of difference equations this method has been also extensively used during the last years (see e.g. [8,10,13,22,28] for difference analogs on the uniform lattice and [3,4,5,8,10,11,14] for the q-case).Later on, references [6, 7, 12] indicated a way of constructing the so-called "dynamical symmetry algebra" by applying the FM to differential or difference equations [2, 10, 11] and then this technique has been used to consider some particular instances of q-hypergeometric difference equations. Of special interest is also the paper by Smirnov [28], in which the equivalence of the FM and the Nikiforov et al formulation of theory of q-orthogonal polynomials [26], was established.…”

“…Of special interest is also the paper by Smirnov [28], in which the equivalence of the FM and the Nikiforov et al formulation of theory of q-orthogonal polynomials [26], was established. In [3], following the papers [14,22] for the classical case, it has been shown that one can factorize the hypergeometric-type difference equation (2.1) in terms of the above-mentioned raising and lowering operators.Our main purpose here is to show how to deal with all different cases of difference equations on the uniform lattice x(s) = s in an unified form. One should consider this paper as an attempt to provide a background for the more general q-linear case (since in the limit as q goes to 1, the q-linear case reduces to the uniform one).…”

“…Of special interest is also the paper by Smirnov [28], in which the equivalence of the FM and the Nikiforov et al formulation of theory of q-orthogonal polynomials [26], was established. In [3], following the papers [14,22] for the classical case, it has been shown that one can factorize the hypergeometric-type difference equation (2.1) in terms of the above-mentioned raising and lowering operators.…”