Matrix Valued Orthogonal Polynomials for Gelfand Pairs of Rank One

Heckman, Gert; Pruijssen, Maarten van

doi:10.48550/arxiv.1310.5134

Cited by 7 publications

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“…However, the focus on the relation with matrix-valued or vector-valued special functions only came later, see e.g. references given in [19], [40]. Grünbaum, Pacharoni, Tirao [17] give a group theoretic approach to matrix-valued orthogonal polynomials emphasising the role of the matrix-valued differential operators, which are manipulated in great detail.…”

Section: Introductionmentioning

confidence: 99%

“…The paper [17] deals with the case (G, K) = (SU(3), U(2)). Motivated by [17] and the approach of Koornwinder [29], the group theoretic interpretation of matrix-valued orthogonal polynomials on (G, K) = (SU(2) × SU(2), SU (2)) is studied from a different point of view, in particular with less manipulation of the matrix-valued differential operators, in [23], [24], see also [19], [40]. The point of view is to construct the matrix-valued orthogonal polynomials using matrix-valued spherical functions, and next using this group theoretic interpretation to obtain properties of the matrixvalued orthogonal polynomials.…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, we know that the Chebyshev polynomials occur as characters on the quantum SU(2) group, see [47, §A.1]. The approach in this paper is to establish the quantum analogue of the group-theoretic approach as presented in [23], [24], see also [19], [40], for the example of the Gelfand pair G = SU(2) × SU (2) with K ∼ = SU (2). For this approach we need Letzter's approach [34], [35], [36] to quantum symmetric spaces using coideal subalgebras.…”

Section: Introductionmentioning

confidence: 99%

“…1.15] on a special case of the branching rule for quantum symmetric pair of type AIII, but in general the lack of the branching rule for the quantum symmetric pairs is an obstacle for the study of quantum analogues of matrix-valued spherical functions of e.g. [19], [17], [40], [38].…”

Section: Introductionmentioning

confidence: 99%

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Matrix-valued orthogonal polynomials related to the quantum analogue of $$(\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})$$ ( SU ( 2 ) × SU ( 2 ) , diag )

2016

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Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of (SU(2) × SU(2), diag) are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey-Wilson polynomials. For these matrix-valued orthogonal polynomials, a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, and the matrix-valued Askey-Wilson type q-difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous q-ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous q-ultraspherical polynomials and q-Racah polynomials. B Pablo Román

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Matrix-valued orthogonal polynomials related to the quantum analogue of $$(\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})$$ ( SU ( 2 ) × SU ( 2 ) , diag )

2016

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“…This was accomplished for the first time in the case of the complex projective plane P 2 (C) = SU(3)/U(2) in [5]. This seminal work gave rise to a series of papers including [6,7,8,10,14,15,16,17,18,19], where one considers matrix valued spherical functions associated to a compact symmetric pair (G, K) of rank one, arriving at sequences of matrix valued orthogonal polynomials of one real variable satisfying an explicit three-term recursion relation, which are also eigenfunctions of a second order matrix differential operator (bispectral property).…”

Section: Introductionmentioning

confidence: 99%

Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere

Tirao

Zurrián

2014

SIGMA

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Abstract. In this paper, we describe the irreducible spherical functions of fundamental K-types associated with the pair (G, K) = (SO(n + 1), SO(n)) in terms of matrix hypergeometric functions. The output of this description is that the irreducible spherical functions of the same K-fundamental type are encoded in new examples of classical sequences of matrix-valued orthogonal polynomials, of size 2 and 3, with respect to a matrix-weight W supported on [0,1]. Moreover, we show that W has a second order symmetric hypergeometric operator D.

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Matrix-Valued Gegenbauer-Type Polynomials

2017

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We introduce matrix-valued weight functions of arbitrary size, which are analogues of the weight function for the Gegenbauer or ultraspherical polynomials for the parameter ν > 0. The LDU-decomposition of the weight is explicitly given in terms of Gegenbauer polynomials. We establish a matrix-valued Pearson equation for these matrix weights leading to explicit shift operators relating the weights for parameters ν and ν + 1. The matrix coefficients of the Pearson equation are obtained using a special matrix-valued differential operator in a commutative algebra of symmetric differential operators. The corresponding orthogonal polynomials Approx (2017) 46:459-487 are the matrix-valued Gegenbauer-type polynomials which are eigenfunctions of the symmetric matrix-valued differential operators. Using the shift operators, we find the squared norm, and we establish a simple Rodrigues formula. The three-term recurrence relation is obtained explicitly using the shift operators as well. We give an explicit nontrivial expression for the matrix entries of the matrix-valued Gegenbauertype polynomials in terms of scalar-valued Gegenbauer and Racah polynomials using the LDU-decomposition and differential operators. The case ν = 1 reduces to the case of matrix-valued Chebyshev polynomials previously obtained using group theoretic considerations.

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Matrix Valued Orthogonal Polynomials for Gelfand Pairs of Rank One

Cited by 7 publications

References 0 publications

Matrix-valued orthogonal polynomials related to the quantum analogue of $$(\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})$$ ( SU ( 2 ) × SU ( 2 ) , diag )

Matrix-valued orthogonal polynomials related to the quantum analogue of $$(\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})$$ ( SU ( 2 ) × SU ( 2 ) , diag )

Spherical Functions of Fundamental K-Types Associated with the n-Dimensional Sphere

Matrix-Valued Gegenbauer-Type Polynomials

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