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Orthogonal polynomials—centroid of their zeroes

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Abstract

Let c n,k (k=1,...,n) the n zeroes of the monic orthogonal polynomials family P n (x). The centroid of these zeroes: \(s_n=\frac1n \sum\limits^n_{k=1}c_{n,k}\) controls globally the distribution of the zeroes, and it is relatively easy to obtain information on s n , like bounds, inequalities, parameters dependence, ..., from the links between s n , the coefficients of the expansion of P n (x), and the coefficients β n , γ n in the basic recurrence relation satisfied by P n (x). After a review of basic properties of the centroid on polynomials, this work gives some results on the centroid of a large class of orthogonal polynomials.

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  1. Department of Mathematics, Université catholique de Louvain, 1348, Louvain-la-Neuve, Belgium

    André Ronveaux

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  1. André Ronveaux
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Correspondence to André Ronveaux.

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In memory of Professor Luigi Gatteschi.

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Ronveaux, A. Orthogonal polynomials—centroid of their zeroes. Numer Algor 49, 373–385 (2008). https://doi.org/10.1007/s11075-008-9195-6

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