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

¹ Department of Mathematics and Applied Mathematics, University of Cape Town 7701, RSA
² Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, RSA

^∗ Corresponding author. E-mail: kathy.driver@uct.ac.za

Abstract

Let {p_n}^∞_n=0 be a sequence of orthogonal polynomials. We briefly review properties of p_n that have been used to derive upper and lower bounds for the largest and smallest zero of p_n. 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.