Interlacing and monotonicity of zeros of Angelesco-Jacobi polynomials

TL;DR

This paper investigates the zeros of Angelesco-Jacobi polynomials, establishing their interlacing and monotonicity properties as parameters vary, and extends these findings to related polynomial families using asymptotic relations.

Contribution

It provides new interlacing and monotonicity results for Angelesco-Jacobi polynomials and extends these properties to multiple Jacobi-Laguerre and Laguerre-Hermite polynomials.

Findings

Zeros of Angelesco-Jacobi polynomials interlace when parameters increase by 1.

Zeros exhibit monotonicity with respect to parameter changes and endpoint variations.

Results are extended to related polynomial families via asymptotic relations.

Abstract

Information about the behavior of zeros of classical families of multiple or Hermite-Pad\'e orthogonal polynomials as functions of the intrinsic parameters of the family is scarce. We establish the interlacing properties of the zeros of Angelesco-Jacobi polynomials when one of the three main parameters is increased by 1, extending the work of dos Santos (2017). We also show their monotonicity with respect to (large values) of the parameter representing in the electrostatic model of the zeros the size of the positive charge fixed at the origin, as well as monotonicity with respect to the endpoint of the interval of orthogonality. These results are extended to zeros of multiple Jacobi-Laguerre and Laguerre-Hermite polynomials using asymptotic relations between these families.