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A survey on orthogonal matrix polynomials satisfying second order differential equations. (English) Zbl 1060.42017

Summary: 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 M. G. Krein devoted some papers to the subject in 1949 [see Am. Math. Soc., Translat., II. Ser. 97, 75–143 (1971); translation from Ukr. Mat. Zh. 1, No. 2, 3–66 (1949; Zbl 0258.47025); Dokl. Akad. Nauk SSSR, n. Ser. 69, 125–128 (1949; Zbl 0035.35904)]. 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.

42C05General theory of orthogonal functions and polynomials