2020/09/20 by Brian Simanek, Simanek, Brian
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Fractional Differential Equations Solutions #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.2009.09432
openalex publication_date 2020/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 to 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.