On the inverse problem of the product of a semi-classical from by a polynomial
A linear functional (or form) \(u\) is called regular if there exists a sequence of polynomials \(\{P_n\}_{n\geq 0}\) with \(\deg P_n=n\) which is orthogonal with respect to \(u\). Such a linear functional is called semi-classical if there exist polynomials \(\Phi\) and \(\Psi\) such that \(D(\Phi u)+\Psi u=0\), where \(D\) denotes the differentiation operator. Under certain conditions, the product of such a semi-classical linear functional and a polynomial is again a semi-classical linear functional. In the paper under review the authors consider the inverse problem : determine all regular linear functionals \(u\) satisfying \(pu=-\lambda v\), where \(p\) is a polynomial, \(\lambda\in\mathbb{C}^*\) and \(v\) is a given regular linear functional. The case that \(p(x)=x^2\) was already considered in [\textit{P. Maroni}, Numer. Algorithms 11, No. 1-4, 243-254 (1996; Zbl 0854.42020)] and in [\textit{J. Alaya} and \textit{P. Maroni}, Methods Appl. Anal. 3, No. 1, 12-30 (1996; Zbl 0865.42022)] in the case that \(v\) is symmetric. In the paper under review the authors consider the case that \(p(x)=x^2\) and \(v\) is nonsymmetric. They derive the structure relation (differential-recurrence relation) of the orthogonal polynomial sequence relative to \(u\). Further they work out an example in detail. In that case they also derive the second order differential equation satisfied by the orthogonal polynomials relative to \(u\) for some special subcases.
scientific article; zbMATH DE number 1145901