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Mathematics > Classical Analysis and ODEs

arXiv:2210.00797 (math)
[Submitted on 3 Oct 2022 (v1), last revised 10 Apr 2023 (this version, v2)]

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

Authors:Alfredo Deaño, Arno B.J. Kuijlaars, Pablo Román
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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 Szegő function. We illustrate the results with two main examples, MVOPs of Jacobi and Gegenbauer type, coming from group theory.
Comments: 48 pages, 8 figures. Some typos corrected in the last version
Subjects: Classical Analysis and ODEs (math.CA)
Cite as: arXiv:2210.00797 [math.CA]
  (or arXiv:2210.00797v2 [math.CA] for this version)
  https://doi.org/10.48550/arXiv.2210.00797

Submission history

From: Alfredo Deaño [view email]
[v1] Mon, 3 Oct 2022 10:10:52 UTC (244 KB)
[v2] Mon, 10 Apr 2023 08:07:56 UTC (252 KB)
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