A survey on orthogonal matrix polynomials satisfying second order differential equations

By A. J. Durán, F. Alberto Grünbaum. 2005

57 citations recorded by the source corpus.

The subject of orthogonal polynomials cuts across a large piece of mathematics and its applications. Two notable examples are mathematical physics in the 19th and 20th centuries, as well as the theory of spherical functions for symmetric spaces. It is also clear that many areas of mathematics grew out of the consideration of problems like the moment problem that are intimately associated to the study of (scalar valued) orthogonal polynomials.Matrix orthogonality on the real line has been sporadically studied during the last half century since Krein devoted some papers to the subject in 1949, see (AMS Translations, Series 2, vol. 97, Providence, Rhode Island, 1971, pp. 75-143, Dokl. Akad. Nauk SSSR 69(2) (1949) 125). In the last decade this study has been made more systematic with the consequence that many basic results of scalar orthogonality have been extended to the matrix case. The most recent of these results is the discovery of important examples of orthogonal matrix polynomials: many families of orthogonal matrix polynomials have been found that (as the classical families of Hermite, Laguerre and Jacobi in the scalar case) satisfy second order differential equations with coefficients independent of n. The aim of this paper is to give an overview of the techniques that have led to these examples, a small sample of the examples themselves and a small step in the challenging direction of finding applications of these new examples.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/9703517eea16b9fd6db28ae234dd8787f04d6d00

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Citations in the corpus

Electrostatic models for zeros of polynomials: Old, new, and some open problems (2005, background): “Duran and Grünbaum gave in [19,20] some examples of matrix orthogonal polynomials { P n } satisfying the second order linear ODE of the form P n (x)A 2 (x) + P n (x)A 1 (x) + P n (x)A 0 (x) = (cid:12) n P n (x), n (cid”
Orthogonal polynomials (2005, unknown): “For all the results in this section see López-Rodriguez–Durán [48] and Durán– Grünbaum [25] and the numerous references there.”
The Analytic Theory of Matrix Orthogonal Polynomials (2007, methodology): “The corresponding real line theory (MOPRL) is still more recent: Following early work of Krein [133] and Berezan’ski [9] on block Jacobi matrices, mainly as applied to self-adjoint extensions, there was a seminal paper of Aptekarev–Nikishin [4] and a flurry of papers since the 1990s [10, 11, 12, 14,”
Orthogonal matrix polynomials satisfying first order differential equations: a collection of instructive examples (2005, methodology): “We observe that the search considered in [6], [ 7 ] deals mainly with cases where the second order differential operator has a scalar as a leading coefficient.”