Second degree semi-classical forms of class s=1. The symmetric case Original Research Article

A form (linear functional) u is called regular if there exists a sequence of polynomials {Pn}n≥0, degPn=n, which is orthogonal with respect to u. Such a form is said to be semi-classical, if there exist two polynomials Φ and Ψ such that (Φu)′+Ψu=0. Now, this form is said to be of second degree if its formal Stieltjes function (see (1.1)) satisfies a second degree equation. Recently, all the second degree classical forms (semi-classical forms of class s=0) are determined. In this paper, we determine all the symmetric semi-classical forms of class s=1, which are also of second degree. Only some forms, introduced by Chihara, which satisfy a certain condition, possess this property. We show that there exists a relation between these forms and the second degree classical ones.