Majorization results for zeros of orthogonal polynomials

TL;DR

This paper establishes linear majorization relations between zeros of consecutive orthogonal polynomials and related polynomials, with explicit doubly stochastic matrices derived from Christoffel numbers, advancing understanding of zero distributions.

Contribution

It introduces explicit doubly stochastic matrices connecting zeros of orthogonal polynomials and their related forms, providing new majorization results.

Findings

Zeros of $p_n$ and $p_{n-1}$ are linearly connected via explicit doubly stochastic matrices.

Similar majorization relations are shown for zeros of $p_n$ and $p_{n-1}^{(1)}$.

Results extend to zeros of polynomials obtained by deleting rows and columns in Jacobi matrices.

Abstract

We show that the zeros of consecutive orthogonal polynomials $p_n$ and $p_{n-1}$ are linearly connected by a doubly stochastic matrix for which the entries are explicitly computed in terms of Christoffel numbers. We give similar results for the zeros of $p_n$ and the associated polynomial $p_{n-1}^{(1)}$ and for the zeros of the polynomial obtained by deleting the $k$th row and column $(1 \leq k \leq n)$ in the corresponding Jacobi matrix.