TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 323, Number I. January 1991 A DISCRETE APPROACH TO MONOTONICITY OF ZEROS OF ORTHOGONAL POLYNOMIALS MOURAD E. H. ISMAIL AND MARTIN E. MULDOON To our friend P. G. Rooney on the occasion of his 65th birthday ABSTRACT. We study the monotonicity with respect to a parameter of zeros of orthogonal polynomials. Our method uses the tridiagonal (Jacobi) matrices arising from the three-term recurrence relation for the polynomials. We obtain new results on monotonicity of zeros of associated Laguerre, Al-Salam-Carlitz, Meixner and PoJlaczek polynomials. We also derive inequalities for the zeros of the Al-Salam-Carlitz and Meixner polynomials. 1. INTRODUCTION The recurrence relation ( 1 ) n = 0, 1, ... , for a system of orthogonal polynomials gives nse to the tridiagonal (Jacobi) matrix Yo Po 0 0 1 15 1 Y 1 PI 0 ... H = [hijJ = 0 15 2 Y 2 P 2 ... J ... The purpose of this work is to present a discrete approach to the question of monotonicity of zeros of a parameter-dependent family of orthogonal polyno- mials as functions of the parameter involved. One may think of the parameter dependence as a perturbation. An outline of our approach will be given in §2. When the coefficients P n , Y n ,t5 n in (1) depend on a parameter r, the zeros of the sn(x) 's will depend also on r and we are particularly interested in the behavior of the zeros as functions of r. The Hellmann-Feynman theorem, The- orem 2.1, gives a formula for the derivative of a zero of SN(X) with respect to r. The definiteness of the derivative of the N x N truncation of H implies the Received by the editors September 12. 1989. 1980 Mathematics Suhject Classification. Primary 33A65; Secondary l5A42. 39A 12. Key words and phrases. Orthogonal polynomials. zeros. monotonicity. recurrence relations. Jacobi matrices. The authors' work was supported by grants from the National Science Foundation and from the Natural Sciences and Engineering Research Council (Canada). 65 © 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use