On semiclassical linear functionals of classs=2: classification and integral representations

Abstract

In this paper, we obtain all the semiclassical linear functionals of class two taking into account the irreducible expression of the corresponding Pearson equation. We focus our attention in their integral representations. Thus, some linear functionals very well known in the literature, associated with perturbations of classical linear functionals, as well as new linear functionals appear which have not been studied as far as we know.

Semi-classical Linear Functionals of Class Four: The Symmetric Case

“…where k is a normalization term and v is the symmetric semi-classical linear form of class two defined by the Pearson equation (see [8,10,17])…”

“…In [1] the classification of symmetric semi-classical linear functionals of such a class is given. Recently, the semi-classical linear functionals of class s = 2 are completely described by F. Marcellán et al in [8,10]. In [9], the authors gives a complete description of all symmetric semi-classical linear of class s = 3 using such an approach.…”

A Classification of Symmetric (1, 1)-Coherent Pairs of Linear Functionals

“…Notice that the classification of semiclassical quasi-definite linear functionals of class s = 1 is given in [13]. The semiclassical linear functionals of Class 2 are described in [41].…”

Discrete semiclassical orthogonal polynomials of class one

“…Examples of semiclassical linear functionals with respect to the derivative operator when the class is either s = 1 (see [9]) or s = 2 (see [35]) have been studied in the literature. Nevertheless, some of them are related to perturbations of classical linear functionals by the addition of Dirac functionals, or their derivatives, supported on convenient points ( [2], [33]).…”