Numerical Algorithms 28: 215–227, 2001.  2001 Kluwer Academic Publishers. Printed in the Netherlands. Symbolic computation of Newton sum rules for the zeros of polynomial eigenfunctions of linear differential operators Tiziana Isoni a , Pierpaolo Natalini b and Paolo E. Ricci a a Dipartimento di Matematica, Università degli Studi di Roma “La Sapienza”, P.le A. Moro, 2-00185 Roma, Italy E-mail: riccip@uniroma1.it b Dipartimento di Matematica, Università degli Studi Roma Tre, Largo San Leonardo Murialdo, 1-00146 Roma, Italy E-mail: natalini@mat.uniroma3.it Received 10 August 2000; revised 2 April 2001 Dedicated to the memory of Prof. Wolf Gross A symbolic algorithm based on the generalized Lucas polynomials of first kind is used in order to compute the Newton sum rules for the zeros of polynomial eigenfunctions of linear differential operators with polynomial coefficients. Keywords: orthogonal polynomials, differential equations with polynomial coefficients, zero’s distribution, Newton sum rules, generalized Lucas polynomials AMS subject classification: 33B15, 33C45, 62E17, 65L99. 1. Introduction The Hamiltonian matrix associated with a dynamical system can be reduced, e.g. by using Householder’s method, to a tridiagonal symmetric (Jacobi) matrix. The char- acteristic polynomials of a chain of principal minors of this matrix constitute a set of orthogonal polynomials. The density of the eigenvalues of the Hamiltonian matrix is then defined as the density distribution of the zeros (shortly: the density of zeros) of such polynomials. By the physical point of view, it is often convenient to deduce information about the density of zeros in terms of the coefficients of the differential operators characterizing the orthogonal polynomial set, without solving explicitly the corresponding differential equations. This permits to analyze the sensitivity of solutions with respect to perturba- tions of the coefficients of the differential operators. The usual analysis of the density distribution of the zeros is performed by study- ing the associated moments, i.e. the normalized Newton sum rules of the zeros itself.