Second degree semi-classical forms of class s=1.: The symmetric case

A form (linear functional) u is called regular if there exists a sequence of polynomials {P-n}(n greater than or equal to 0), deg P-n = n, which is orthogonal with respect to u. Such a form is said to be semi-classical, if there exist two polynomials Phi and psi such that (Phi u)' + psi u = 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. (C) 2000 IMACS. Published by Elsevier Science B.V. All rights reserved.