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Perturbed zeros of classical orthogonal polynomials. (English) Zbl 0958.33005
It is known that the set of N zeros of polynomials satisfying linear homogeneous second order differential equations is the solution of certain system of N nonlinear equations depending on a function related to one of the coefficients of the differential equation and, in the case of the classical orthogonal polynomials to the corresponding weight function. By perturbing such a function the author generates sets of perturbed zeros. This procedure yields a linear problem for the differences between the original and perturbed zeros. The matrix of this linear problem is the one used by Stieltjes to show the monotonic variation of the zeros of the classical orthogonal polynomials. The elementary method presented in this paper can be used to give a bound for the error produced in approximating the original set of zeros by the perturbed one. As simple examples the author obtains a bound for the euclidean norm of the vector of differences between the zeros of Hermite polynomials and those of (approximately scaled) Gegenbauer and factorial polynomials.
MSC:
33C45Orthogonal polynomials and functions of hypergeometric type
15A45Miscellaneous inequalities involving matrices