Behavior of zeros of $X_{1}$-Jacobi and $X_{1}$-Laguerre exceptional polynomials

TL;DR

This paper investigates the zeros of $X_1$-Jacobi and $X_1$-Laguerre exceptional orthogonal polynomials, focusing on their interlacing, monotonicity, and parameter dependence.

Contribution

It establishes new properties of the zeros, including interlacing and monotonicity, for these specific classes of exceptional orthogonal polynomials.

Findings

Zeros exhibit interlacing properties.

Zeros show monotonicity with respect to parameters.

Distinct behaviors of regular and exceptional zeros identified.

Abstract

The $X_1$-Jacobi and the $X_1$-Laguerre exceptional orthogonal polynomials have been introduced and studied by G\'omez-Ullate, Kamran and Milson in a series of papers. In this note, we establish some properties, such as interlacing, monotonicity with respect to the parameters and order, about the so-called \textit{regular} and \textit{exceptional} zeros of these two classes of polynomials.