The Algebra of Differential Operators Associated to a Weight Matrix

Grünbaum, F. Alberto; Tirao, Juan

doi:10.1007/s00020-007-1517-x

Cited by 62 publications

(73 citation statements)

References 33 publications

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“…The symmetry of the operators with respect to W means that P D, Q W = P, QD W and P E, Q W = P, QE W for all matrix-valued polynomials with respect to the matrix-valued inner product ·, · W defined in (1.2). The last statement follows immediately from the first by the results of Grünbaum and Tirao [10]. Also, [D, E] = 0 follows from the fact that the eigenvalue matrices commute.…”

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confidence: 48%

“…For more information on differential operators for matrix-valued functions, see e.g. [10], [21].We denote by E ij the standard matrix units, i.e. E ij is the matrix with all matrix entries equal to zero, except for the (i, j)-th entry which is 1.…”

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confidence: 99%

“…By general considerations, e.g. [10], we can construct the corresponding monic matrix-valued orthogonal polynomials {P n } ∞ n=0 , sowhere H n > 0 means that H n is a positive definite matrix, P n (x) = n k=0 x k P n k with P n k ∈ M 2ℓ+1 (C) and P n n = I, the identity matrix. The polynomials P n are the monic variants of the matrix-valued orthogonal polynomials constructed in [15] from representation theoretic considerations.…”

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confidence: 99%

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Matrix-valued Orthogonal Polynomials Related to (SU(2)$ \times$ SU(2), diag), II

Koelink¹,

Pruijssen²,

Román³

2013

Publ. Res. Inst. Math. Sci.

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Abstract. In a previous paper we have introduced matrix-valued analogues of the Chebyshev polynomials by studying matrix-valued spherical functions on SU(2) × SU(2). In particular the matrix-size of the polynomials is arbitrarily large. The matrix-valued orthogonal polynomials and the corresponding weight function are studied. In particular, we calculate the LDU-decomposition of the weight where the matrix entries of L are given in terms of Gegenbauer polynomials. The monic matrix-valued orthogonal polynomials P n are expressed in terms of Tirao's matrix-valued hypergeometric function using the matrix-valued differential operator of first and second order to which the P n 's are eigenfunctions. From this result we obtain an explicit formula for coefficients in the three-term recurrence relation satisfied by the polynomials P n . These differential operators are also crucial in expressing the matrix entries of P n L as a product of a Racah and a Gegenbauer polynomial. We also present a group theoretic derivation of the matrix-valued differential operators by considering the Casimir operators corresponding to SU(2) × SU(2). IntroductionMatrix-valued orthogonal polynomials have been studied from different perspectives in recent years. Originally they have been introduced by Krein [17], [18]. Matrix-valued orthogonal polynomials have been related to various different subjects, such as higher-order recurrence equations, spectral decompositions, and representation theory. The matrix-valued orthogonal polynomials studied in this paper arise from the representation theory of the group SU(2) × SU(2) with the compact subgroup SU(2) embedded diagonally, see [15] for this particular case and Gangolli and Varadarajan [8], Tirao [20], Warner [22] for general group theoretic interpretations of matrix-valued spherical functions. An important example is the study of the matrix-valued orthogonal polynomials for the case (SU(3), U(2)), which has been studied by Grünbaum, Pacharoni and Tirao [9] mainly exploiting the invariant differential operators. In [15] we have studied the matrix-valued orthogonal operators related to the case (SU(2) × SU(2), SU(2)), which lead to the matrix-valued orthogonal polynomial analogues of Chebyshev polynomials of the second kind U n , in a different fashion. In the current paper we study these matrix-valued orthogonal polynomials in more detail. N, n, m ∈ {0, 1, · · · , 2ℓ}, and U n is the Chebyshev polynomial of the second kind. Note that the sum in (1.1) actually starts at min(0, n + m − 2ℓ). It follows that W :, is a (2ℓ+1)×(2ℓ+1)-matrix-valued integrable function such that all moments 1 −1 x n W (x) dx, n ∈ N, exist. From the construction given in [15, §5] it follows W (x) is positive definite almost everywhere. By general considerations, e.g. [10], we can construct the corresponding monic matrix-valued orthogonal polynomials {P n } ∞ n=0 , sowhere H n > 0 means that H n is a positive definite matrix, P n (x) = n k=0 x k P n k with P n k ∈ M 2ℓ+1 (C) and P n n = I, the identity matrix. The ...

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confidence: 48%

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confidence: 99%

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confidence: 99%

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Matrix-valued Orthogonal Polynomials Related to (SU(2)$ \times$ SU(2), diag), II

Koelink¹,

Pruijssen²,

Román³

2013

Publ. Res. Inst. Math. Sci.

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“…General nonorthogonal decompositions are governed by the real vector space [22,44]. In [28] we show that A (ν) equals the Hermitian elements of A (ν) , so that there is no further nonorthogonal decomposition.…”

Section: Proposition 26mentioning

confidence: 91%

“…Note that H (ν) 0 can be calculated using the explicit expression of Definition 2.1 and the orthogonality relations (1.3), which gives the special case n = 0 of Theorem 3.1(i). In case Q n is another set of matrixvalued orthogonal polynomials with respect to the weight function W (ν) on (−1, 1), then there exist invertible matrices E n so that Q n (x) = E n P n (x) for all x and all n, see [13,22].…”

Section: The Matrix-valued Gegenbauer-type Polynomials and Their Propmentioning

confidence: 99%

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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The Darboux process and a noncommutative bispectral problem: some explorations and challenges

Grünbaum

2011

Progress in Mathematics

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The Darboux process, also known by many other names, played a very important role in some extremely enjoyable joint work that Hans and I did 25 years ago. I revisit a version of this problem in a case when scalars are replaced by matrices, i.e., elements of a non-commutative ring. Many of the issues studied here can be pushed to the case of a ring with identity, but my emphasis is on very concrete examples involving 2 × 2 matrices.2000 Mathematics Subject Classification. 33C45, 22E45.

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The Algebra of Differential Operators Associated to a Weight Matrix

Cited by 62 publications

References 33 publications

Matrix-valued Orthogonal Polynomials Related to (SU(2)$ \times$ SU(2), diag), II

Matrix-valued Orthogonal Polynomials Related to (SU(2)$ \times$ SU(2), diag), II

Matrix-Valued Gegenbauer-Type Polynomials

The Darboux process and a noncommutative bispectral problem: some explorations and challenges

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