On another characterization of Askey-Wilson polynomials

Abstract

In this paper we show that the only sequences of orthogonal polynomials (Pn) n≥0 satisfying(cn = 0) where φ is a well chosen polynomial of degree at most two, Dq is the Askey-Wilson operator and Sq the averaging operator, are the multiple of Askey-Wilson polynomials, or specific or limiting cases of them.

Characterization of Orthogonal Polynomials on lattices

“…That is to characterize all orthogonal polynomials sequences (OPS), (P n ) n≥0 , solutions of the following equation (az 2 + bz + c) △ △x(s − 1/2) P n (x(s − 1/2)) (1.1) = (△ + 2 I)(a n P n+1 + b n P n + c n P n−1 )(x(s − 1/2)), where I is the identity operator, a, b and c are some well chosen complex numbers, x defines a class of lattices (or grids) with, generally, nonuniform step-size, △f (s) = f (s + 1) − f (s), and ∇f (s) = △f (s − 1). The case where the lattice x is qquadratic and given by x(s) = (q −s + q s )/2 was solved recently in [12], where under some conditions imposed in a, b and c, the only solutions are the Askey-Wilson polynomials including special or limiting cases of them. But as noticed in [9], when we consider a quadratic lattice for (1.1), solutions "can not easily be deduced from those of Askey-Wilson polynomials".…”

“…In addition it is proved in [12] that the following equation has classical OPS as solutions b n,j △ △x(s − 1/2) P j (x(s − 1/2)) .…”