%0 Journal Article
%T On the Complex Zeros of Some Families of Orthogonal Polynomials
%A Eugenia N. Petropoulou
%J Abstract and Applied Analysis
%D 2010
%I Hindawi Publishing Corporation
%R 10.1155/2010/263860
%X The complex zeros of the orthogonal Laguerre polynomials for , ultraspherical polynomials for , Jacobi polynomials for , , , orthonormal Al-Salam-Carlitz II polynomials for , , and -Laguerre polynomials for , are studied. Several inequalities regarding the real and imaginary properties of these zeros are given, which help locating their position. Moreover, a few limit relations regarding the asymptotic behavior of these zeros are proved. The method used is a functional analytic one. The obtained results complement and improve previously known results. 1. Introduction Orthogonal polynomials appear naturally in various problems of physics and mathematics and are considered as one of the basic tools in confronting problems of mathematical physics. Also, orthogonal polynomials have many important applications in problems of numerical analysis, such as interpolation or optimization. For a survey on applications and computational aspects of orthogonal polynomials, see [1] and the references therein. Some of the most important properties of orthogonal polynomials, , are the following. (P1)The orthogonal polynomials are orthogonal with respect to a weight function on an interval of orthogonality and all their zeros are real and simple and lie inside . (P2)Some classes of orthogonal polynomials (including some of the classes studied in the present paper) satisfy an ordinary differential equation of the form where is a polynomial of degree at most two, is a polynomial of degree exactly one, and is a constant. (P3)The orthogonal polynomials satisfy a three-term recurrence relation of the form where . An analog to the theory of classical orthogonal polynomials has recently been developed for -polynomials, , which also appear in various areas of mathematics and physics. The -polynomials satisfy also a recurrence relation of the form (1.2), but now the sequences , and as well as the polynomials depend on the parameter . On the other hand the -polynomials do not satisfy a differential equation, but a -difference equation which is considered as the -analog of (1.1). For more information on classical or -polynomials one may consult [2¨C6] and the references therein. Also, -polynomials arise in the context of indeterminate moment problems. In this case, there are some classes of orthogonal polynomials for which the corresponding measure of orthogonality is not unique. This may give rise to various types of -polynomials, other than the ones studied in the present paper. For more information see [4, 7¨C9] and the references therein. Due to their importance, orthogonal
%U http://www.hindawi.com/journals/aaa/2010/263860/