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

• Published in: Integral transforms and special functions Vol. 34; no. 1; pp. 65 - 87
• Main Authors: Marcellán, Francisco, Khalfallah, Mohamed
• Format: Journal Article
• Language: English
• Published: Abingdon Taylor & Francis 02-01-2023; Taylor & Francis Ltd
• Subjects: Classical and semiclassical forms; Orthogonal polynomials; Polynomial mappings; Polynomials; Primary: 33C45; Secondary: 42C05; Stieltjes function; Third-degree forms
• ISSN: 1065-2469; 1476-8291
• DOI: 10.1080/10652469.2022.2089134

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.