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Interlacing and monotonicity of zeros of Angelesco-Jacobi polynomials

Andrei Martinez-Finkelshtein, Rafael Morales

Abstract

Information about the behavior of zeros of classical families of multiple or Hermite-Padé 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.

Interlacing and monotonicity of zeros of Angelesco-Jacobi polynomials

Abstract

Information about the behavior of zeros of classical families of multiple or Hermite-Padé 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.
Paper Structure (7 sections, 16 theorems, 116 equations, 1 figure)

This paper contains 7 sections, 16 theorems, 116 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof of Theorem \ref{['entrelazamiento']}
  4. Mean behavior of the zeros
  5. Proof of Theorem \ref{['thm:monotonicity']}
  6. Proof of Theorem \ref{['thm:monotonicityA']}
  7. Proof of Theorem \ref{['thm:interlacingLaguerre']}

Key Result

Theorem A

Given $n\in\mathbb{N}$, the zeros zerosJacobi of $P_n^{(\alpha,\beta)}(x)$ decrease with respect to $\alpha$ and increase with respect to $\beta$.

Figures (1)

  • Figure 1: Graph of the polynomials $P_5^{(-1/2,1,0)}(x;-2)$ (dashed line) and $P_4^{(1/2,2,1)}(x;-2)$ (thick line) on $[-2,1]$.

Theorems & Definitions (31)

  • Theorem A
  • Definition 1.1: Interlacing
  • Theorem B
  • Theorem C
  • Remark 2.1
  • Theorem 2.2
  • Theorem D
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • ...and 21 more