A survey on orthogonal matrix polynomials satisfying second order differential equations

Abstract. The main purpose of this paper is to obtain an explicit expression of a family of matrix valued orthogonal polynomials {Pn}n, with respect to a weight W , that are eigenfunctions of a second order differential operator D. The weight W and the differential operator D were found in [12], using some aspects of the theory of the spherical functions associated to the complex projective spaces. We also find other second order differential operator E symmetric with respect to W and we describe the algebra generated by D and E.

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“…In [8] (see also [3]) it is proved that the condition of symmetry for D is equivalent to the following three differential equations…”

Section: Introductionmentioning

confidence: 99%

A Sequence of Matrix Valued Orthogonal Polynomials Associated to Spherical Functions

Pacharoni

Román

2007

Constr Approx

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“…Theorem 2.2 is a particular case of Theorem 2.3 of [D3] (the special case when F 2 is a scalar function is Theorem 4.1 of [DG1]). …”

Section: Preliminariesmentioning

confidence: 99%

“…F 1 (t) + F 0 (t) (1.1) has been introduced (see [DG1], [DG3], [D2], [D3], [DdI1], [DdI2], [G], [GPT1], [GPT2], [CMV], [PT]). The differential coefficients F 2 , F 1 and F 0 are matrix polynomials (which do not depend on n) of degrees less than or equal to 2, 1 and 0, respectively.…”

Section: Introductionmentioning

confidence: 99%

Orthogonal matrix polynomials satisfying differential equations with recurrence coefficients having non-scalar limits

Borrego

Castro

Durán

2012

Integral Transforms and Special Functions

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Abstract. We introduce a family of weight matrices W of the form T (t)T * (t), T (t) = e A t e Dt 2 , where A is certain nilpotent matrix and D is a diagonal matrix with negative real entries. The weight matrices W have arbitrary size N × N and depend on N parameters. The orthogonal polynomials with respect to this family of weight matrices satisfy a second order differential equation with differential coefficients that are matrix polynomials F 2 , F 1 and F 0 (independent of n) of degrees not bigger than 2, 1 and 0 respectively. For size 2 × 2, we find an explicit expression for a sequence of orthonormal polynomials with respect to W . In particular, we show that one of the recurrence coefficients for this sequence of orthonormal polynomials does not asymptotically behave as a scalar multiple of the identity, as it happens in the examples studied up to now in the literature.

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“…Tirao's work on the theory of matrix-valued spherical functions, see [Tir77], one finds in [GPT02a,GPT03,Grü03] the first examples of situations as those considered in [Dur97]; in [DG04] one can find the development of a general method that leads to introduce and study classes of examples of orthonormal matrix polynomials. In the last years, one can see a growing number of papers devoted to different aspects of this question.…”

Section: Introductionmentioning

confidence: 99%

The Algebra of Differential Operators for a Matrix Weight: An Ultraspherical Example

Zurrián¹

2016

Int Math Res Notices

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Abstract. In this paper we study in detail algebraic properties of the algebra D(W ) of differential operators associated to a matrix weight of Gegenbauer type. We prove that two second order operators generate the algebra, indeed D(W ) is isomorphic to the free algebra generated by two elements subject to certain relations. Also, the center is isomorphic to the affine algebra of a singular rational curve. The algebra D(W ) is a finitely-generated torsion-free module over its center, but it is not flat and therefore it is not projective. This is the second detailed study of an algebra D(W ) and the first one coming from spherical functions and group representations. We prove that the algebras for different Gegenbauer weights and the algebras studied previously, related to Hermite weights, are isomorphic to each other. We give some general results that allow us to regard the algebra D(W ) as the centralizer of its center in the Weyl algebra. We do believe that this should hold for any irreducible weight and the case considered in this paper represents a good step in this direction.

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A survey on orthogonal matrix polynomials satisfying second order differential equations

Cited by 86 publications

References 35 publications

A Sequence of Matrix Valued Orthogonal Polynomials Associated to Spherical Functions

A Sequence of Matrix Valued Orthogonal Polynomials Associated to Spherical Functions

Orthogonal matrix polynomials satisfying differential equations with recurrence coefficients having non-scalar limits

The Algebra of Differential Operators for a Matrix Weight: An Ultraspherical Example

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