Asymptotic behaviour of zeros of exceptional Jacobi and Laguerre polynomials

TL;DR

This paper analyzes the asymptotic behavior and zero distribution of exceptional Jacobi and Laguerre polynomials, revealing interlacing properties and providing a generalized Heine-Mehler formula.

Contribution

It introduces a generalized Heine-Mehler formula for exceptional polynomials and characterizes the asymptotic zero distribution, including the convergence of exceptional zeros.

Findings

Regular zeros interlace with classical zeros

Exceptional zeros converge to fixed points as n grows

A generalized Heine-Mehler formula is established

Abstract

The location and asymptotic behaviour for large n of the zeros of exceptional Jacobi and Laguerre polynomials are discussed. The zeros of exceptional polynomials fall into two classes: the regular zeros, which lie in the interval of orthogonality and the exceptional zeros, which lie outside that interval. We show that the regular zeros have two interlacing properties: one is the natural interlacing between consecutive polynomials as a consequence of their Sturm-Liouville character, while the other one shows interlacing between the zeros of exceptional and classical polynomials. A generalization of the classical Heine-Mehler formula is provided for the exceptional polynomials, which allows to derive the asymptotic behaviour of their regular zeros. We also describe the location and the asymptotic behaviour of the exceptional zeros, which converge for large n to fixed values.