On the functional equation for classical orthogonal polynomials on lattices

Some Appell-type orthogonal polynomials on lattices

“…Remark 4.1. Although the results obtained here were proved for the q-quadratic lattices, they can be easily extended to quadratic lattices x(s) = c 4 s 2 + c 5 s + c 6 by taking the appropriate limit as it was discussed in [10].…”

A structure relation for some specific orthogonal polynomials

“…Then (P n ) n≥0 satisfies the following other relation (ax − c)U 2 (x)D q P n (x) = r [1] n P n+2 (x) + r [2] n P n+1 (x) + r [3] n P n (x) (2.6) + r [4] n P n−1 (x) + r [5] n P n−2 (x) , for each n = 0, 1, 2, . .…”

“…The approach with lattices was most welcome because this is useful to describe in an unified way families of orthogonal polynomial sequences (OPS) including classical ones. For a recent reference on the subject we refer the reader to [3] including some reference therein, where some properties of the so-called Askey-Wilson operator and Askey-Wilson polynomials are studied. Despite the fact that classical OPS (on lattices) constitute the most studied class of OPS, they are still some interesting unsolved problems (see [5, p. 653]).…”

Remarks on the paper "Recurrence equations and their classical orthogonal polynomial solutions on a quadratic or a q-quadratic lattice"

“…This also reveals some asymptotic behaviour and relation between coefficients of the TTRR of such OPS. We focus only on classical OPS with respect to the Askey-Wilson operator (so on q-quadratic lattices) since, from this, we can obtained similar results for classical OPS on quadratic lattices as discussed in [4].…”