On the characterizations of third-degree semiclassical forms via polynomial mappings

Abstract

The aim of this contribution is the study of orthogonal polynomials via polynomial mappings in the framework of the third-degree semiclassical linear forms. Let u and v be two regular forms and let denote by and the corresponding sequences of monic orthogonal polynomials such that there exists a monic polynomial of degree m, with and in such a way where k is a fixed integer number such that If u (resp. v) is a third-degree linear form, then we prove that the other one is also a third-degree linear form. From this fact we are able to show the relation between third-degree semiclassical forms u of class and the classical forms. More precisely, the strict third-degree (respectively second-degree) forms are rational modifications of the product of k shifted Jacobi forms (resp. ). An illustrative example is given.