Asymptotics of matrix valued orthogonal polynomials on $[-1,1]$

TL;DR

This paper studies the large degree asymptotic behavior of matrix valued orthogonal polynomials with Jacobi-type weights, extending scalar techniques to matrix cases and providing detailed asymptotic expansions in different regions.

Contribution

It develops a Riemann-Hilbert analysis for MVOPs with matrix weights, including factorizations and asymptotic expansions, extending scalar orthogonal polynomial methods to the matrix setting.

Findings

Derived asymptotic expansions for MVOPs in various regions of the complex plane.

Provided explicit examples for Jacobi and Gegenbauer type MVOPs from group theory.

Extended scalar orthogonal polynomial techniques to matrix-valued cases.

Abstract

We analyze the large degree asymptotic behavior of matrix valued orthogonal polynomials (MVOPs), with a weight that consists of a Jacobi scalar factor and a matrix part. Using the Riemann-Hilbert formulation for MVOPs and the Deift-Zhou method of steepest descent, we obtain asymptotic expansions for the MVOPs as the degree tends to infinity, in different regions of the complex plane (outside the interval of orthogonality, on the interval away from the endpoints and in neighborhoods of the endpoints), as well as for the matrix coefficients in the three-term recurrence relation for these MVOPs. The asymptotic analysis follows the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen on scalar Jacobi-type orthogonal polynomials, but it also requires several different factorizations of the matrix part of the weight, in terms of eigenvalues/eigenvectors and using a matrix Szeg\H{o} function. We illustrate the results with two main examples, MVOPs of Jacobi and Gegenbauer type, coming from group theory.