Regular Article
The Asymptotic Zero Distribution of Orthogonal Polynomials with Varying Recurrence Coefficients

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Abstract

We study the zeros of orthogonal polynomials pnN, n=0, 1, …, that are generated by recurrence coefficients anN and bnN depending on a parameter N. Assuming that the recurrence coefficients converge whenever nN tend to infinity in such a way that the ratio n/N converges, we show that the polynomials pnN have an asymptotic zero distribution as n/N tends to t>0 and we present an explicit formula for the limiting measure. This formula contains the asymptotic zero distri- butions for various special classes of orthogonal polynomials that were found earlier by different methods, such as Jacobi polynomials with varying parameters, discrete Chebyshev polynomials, Krawtchouk polynomials, and Tricomi–Carlitz polynomials. We also give new results on zero distributions of Charlier polynomials, Stieltjes–Wigert polynomials, and Lommel polynomials.

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Communicated by Guillermo, López Lagomasino

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Current address: Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium. E-mail: [email protected].

The second author is a Research Director of the National Fund for Scientific Research (FWO). This work is partially supported by the FWO research project G.0278.97 and INTAS project 93-129ext.

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E-mail: [email protected]