A characterization theorem for semi-classical orthogonal polynomials on non-uniform lattices

Abstract

It is proved a characterization theorem for semi-classical orthogonal polynomials on non-uniform lattices that states the equivalence between the Pearson equation for the weight and some systems involving the orthogonal polynomials as well as the functions of the second kind. As a consequence, it is deduced the analogue of the so-called compatibility conditions in the ladder operator scheme. The classical orthogonal polynomials on non-uniform lattices are then recovered under such compatibility conditions, through a closed formula for the recurrence coefficients.

Discrete semi-classical orthogonal polynomials of class one on quadratic lattices

“…Further matrix identities. The following results extend the differential systems from the continuous orthogonality given in [14] to the discrete orthogonality on systems of nonuniform lattices (see [3,Th. 1] and also [22,Sec.…”

“…(1) Here, D is some divided-difference operator and M is a companion difference operator related to D. The divided-difference calculus is classified in terms of hierarchies of operators and related lattices (see, for instance, [22,Sec. 2,3]). In this paper we shall consider the divided-difference operator D given by There are many papers on semi-classical orthogonal polynomials on quadratic lattices.…”

On the second-order holonomic equation for Sobolev-type orthogonal polynomials

“…with A, C, D irreducible polynomials (in x). Furthermore, whenever S is defined through (22), with µ defined in terms of a weight w as dµ(x) = w(x)dx, then we also have the equivalence between (i) and (ii) and the Pearson equation for the weight [8,27],…”

“…6] and [21,Sec. 4.2] using different aproaches/techniques than the ones in the present paper (here, we take advantage of some of difference systems deduced in [8]). Also, let us note that standard techniques, such as the ladder-operator approach (see, for for instance, [2] or [16,Sec.…”