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Convergence Rates of Exceptional Zeros of Exceptional Orthogonal Polynomials

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Abstract

We consider the zeros of exceptional orthogonal polynomials (XOP). Exceptional orthogonal polynomials were originally discovered as eigenfunctions of second order differential operators that exist outside the classical Bochner–Brenke classification due to the fact that XOP sequences omit polynomials of certain degrees. This omission causes several properties of the classical orthogonal polynomial sequences to not extend to the XOP sequences. One such property is the restriction of the zeros to the convex hull of the support of the measure of orthogonality. In the XOP case, the zeros that exist outside the classical intervals are called exceptional zeros and they often converge toward easily identifiable limit points as the degree becomes large. We deduce the exact rate of convergence and verify that certain estimates that previously appeared in the literature are sharp.

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Acknowledgements

The author is grateful to the anonymous referee for many useful suggestions.

Funding

The author gratefully acknowledges support from the Simons Foundation through collaboration grant 707882.

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  1. Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX, 76798, USA

    Brian Simanek

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  1. Brian Simanek
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Correspondence to Brian Simanek.

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Communicated by Laurent Baratchart.

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Simanek, B. Convergence Rates of Exceptional Zeros of Exceptional Orthogonal Polynomials. Comput. Methods Funct. Theory 23, 629–649 (2023). https://doi.org/10.1007/s40315-022-00459-z

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