Factorization of the hypergeometric-type difference equation on non-uniform lattices: dynamical algebra. (English) Zbl 1079.33017

Álvarez-Nodarse, R.; Atakishiyev, N. M.; Costas-Santos, R. S.

J. Phys. A, Math. Gen. 38, No. 1, 53-174 (2005).

This paper is a continuation of the study started in [R. Alvarez-Nodarse and R. S. Costas-Santos, J. Phys. A, Math. Gen. 34, 5551–5569 (2001; Zbl 0994.39001)]. The authors study the factorization of a second-order \(q\)-difference operator into a so-called raising and lowering operator. It is shown that in most cases of \(q\)-hypergeometric type this facorization leads to the dynamical algebra \(\text{su}_q(1,1)\) whose generators are explicitly constructed in terms of the difference operators obtained in the process of factorization. The general results are illustrated by a number of explicit examples of \(q\)-hypergeometric orthogonal polynomials.

Reviewer: Roelof Koekoek (Delft)

MSC:

33E30

Other functions coming from differential, difference and integral equations

Keywords:

\(q\)-difference equation; \(q\)-orthogonal polynomials

Citations:

Zbl 0994.39001