Semiclassical Forms of Class s = 2: The Symmetric Case, when \Phi(0) = 0

Abstract

A regular linear form u is said to be semiclassical, if there exist two polynomials  monic and , deg( )  1 such that (u)' + u = 0. Recently, all the symmetric semiclassical linear forms of class s  1 are determined. In this paper, by considering the inverse problem of the product of a form by a polynomial in the square case, we carry out the complete description of the symmetric semiclassical linear forms of class s = 2, when (0) = 0 which generalize those of class s = 1. Essentially, three canonical cases appear. Some particular cases refer to well-known orthogonal sequences. Representations of these linear forms are given.