Inequalities for Extreme Zeros of Some Classical Orthogonal and q-orthogonal Polynomials

Let {pn}1 n=0 be a sequence of orthogonal polynomials. We briefly review properties of pn that have been used to derive upper and lower bounds for the largest and smallest zero of pn. Bounds for the extreme zeros of Laguerre, Jacobi and Gegenbauer polynomials that have been obtained using different approaches are numerically compared and new bounds for extreme zeros of q-Laguerre and little q-Jacobi polynomials are proved

Reliable Computation of the Zeros of Solutions of Second Order Linear ODEs Using a Fourth Order Method

A fourth order fixed point method to compute the zeros of solutions of second order homogeneous linear ODEs is obtained from the approximate integration of the Riccati equation associated with the ODE. The method requires the evaluation of the logarithmic derivative of the function and also uses the coefficients of the ODE. An algorithm to compute with certainty all the zeros in an interval is given which provides a fast, reliable, and accurate method of computation. The method is illustrated by the computation of the zeros of Gauss hypergeometric functions (including Jacobi polynomials) and confluent hypergeometric functions (Laguerre polynomials, Hermite polynomials, and Bessel functions included) among others. The examples show that typically 4 or 5 iterations per root are enough to provide more than 100 digits of accuracy, without requiring a priori estimations of the roots

Asymptotics of the L^2 Norm of Derivatives of OPUC

We show that for many families of OPUC, one has $||\varphi'_n||_2/n -> 1$, a condition we call normal behavior. We prove that this implies $|\alpha_n| -> 0$ and that it holds if the sequence $\alpha_n$ is in $\ell^1$. We also prove it is true for many sparse sequences. On the other hand, it is often destroyed by the insertion of a mass point.Comment: 36 pages, no figures. Minor corrections, to appear in the Journal of Approximation Theor

On co-polynomials on the real line

In this paper, we study new algebraic and analytic aspects of orthogonal polynomials on the real line when finite modifications of the recurrence coefficients, the so-called co-polynomials on the real line, are considered. We investigate the behavior of their zeros, mainly interlacing and monotonicity properties. Furthermore, using a transfer matrix approach we obtain new structural relations, combining theoretical and computational advantages. Finally, a connection with the theory of orthogonal polynomials on the unit circle is pointed out.The authors thank the referee for constructive comments and recommendations which improved the readability and quality of the manuscript. The research of the first author is supported by the Portuguese Government through the Fundação para a Ciência e a Tecnologia (FCT) under the grant SFRH/BPD/ 101139/2014. This author also acknowledges the financial support by the Brazilian Government through the CNPq under the project 470019/2013-1. The research of the first and second author is supported by the Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain under the project MTM2012-36732-C03-01. The second author also acknowledges the financial support by the Brazilian Government through the CAPES under the project 107/2012

Electrostatic models for zeros of polynomials: Old, new, and some open problems

15 pages, 2 figures.-- MSC2000 codes: Primary, 30C15; Secondary, 34C10; 33C45; 42C05; 82B23.MR#: MR2345246 (2008h:33017)Zbl#: Zbl 1131.30002We give a survey concerning both very classical and recent results on the electrostatic interpretation of the zeros of some well-known families of polynomials, and the interplay between these models and the asymptotic distribution of their zeros when the degree of the polynomials tends to infinity. The leading role is played by the differential equation satisfied by these polynomials. Some new developments, applications and open problems are presented.This research was supported, in part, by a grant of Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain, project code BFM2003-06335-C03-02 (FM), a grant from the Ministry of Education and Science of Spain, project code MTM2005-08648-C02-01 (AMF, PMG), by Junta de Andalucía, Grupo de Investigación FQM 0229 (AMF, PMG), by “Research Network on Constructive Complex Approximation (NeCCA)”, INTAS 03-51-6637 (FM, AMF), and by NATO Collaborative Linkage Grant “Orthogonal Polynomials: Theory, Applications and Generalizations”, ref. PST.CLG.979738 (FM, AMF).Publicad