Convergence Rates of Exceptional Zeros of Exceptional Orthogonal Polynomials

TL;DR

This paper investigates the convergence rates of exceptional zeros of exceptional orthogonal polynomials, providing precise estimates and confirming the sharpness of previous bounds, thus advancing understanding of their asymptotic behavior.

Contribution

It derives exact convergence rates for exceptional zeros of XOP and confirms the sharpness of existing estimates, extending classical orthogonal polynomial theory.

Findings

Exceptional zeros converge to specific limit points.

Derived exact convergence rates for exceptional zeros.

Validated sharpness of previously proposed estimates.

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 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.