Elementary examples of solutions to Bochner’s problem for matrix differential operators

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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“…Darboux transformations for matrix-valued differential operators require more study, see, e.g., [4,11,12,21].…”

Section: Qs (ν) (ν) For Matrix-valued Functions P and Q With Cmentioning

confidence: 99%

Matrix-Valued Gegenbauer-Type Polynomials

2017

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“…The study of the structure of the algebra of possible θ(x) going with a fixed bispectral ψ(x, z) was first raised in Castro and Grünbaum (2006) and analyzed in Tirao (2011); Grünbaum and Tirao (2007). See also Casper (2018) and Zurrián (2017).…”

Section: The Non-commutative Version Of the Bispectral Problemmentioning

confidence: 99%

Matrix Bispectrality and Noncommutative Algebras: beyond the prolate spheroidals

Grünbaum¹,

Vasquez²,

Zubelli³

2022

Preprint

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“…Remark 7. 6 In [42], the authors study the algebra D W ( p,q) , where W ( p,q) is, for p = q 2 , the irreducible weight matrix…”

Section: The Algebra D (W)mentioning

confidence: 99%

“…7, we describe the algebra of second-order differential operators associated with the weight matrix W (α,β,v) given in (2.4) and (2.5). Indeed, for a given weight matrix W , the analysis of the algebra D(W ) of all differential operators that have a sequence of matrix-valued orthogonal polynomials with respect to W as eigenfunctions has received much attention in the literature in the last fifteen years [6,8,9,33,42,45,47]. While for classical orthogonal polynomials, the structure of this algebra is very well-known (see [39]), in the matrix setting, where this algebra is non-commutative, the situation is highly non-trivial.…”

Section: Introductionmentioning

confidence: 99%

Structural Formulas for Matrix-Valued Orthogonal Polynomials Related to $$2\times 2$$ Hypergeometric Operators

Calderón

Castro

2021

Bull. Malays. Math. Sci. Soc.

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We give some structural formulas for the family of matrix-valued orthogonal polynomials of size $$2\times 2$$ 2 × 2 introduced by C. Calderón et al. in an earlier work, which are common eigenfunctions of a differential operator of hypergeometric type. Specifically, we give a Rodrigues formula that allows us to write this family of polynomials explicitly in terms of the classical Jacobi polynomials, and write, for the sequence of orthonormal polynomials, the three-term recurrence relation and the Christoffel–Darboux identity. We obtain a Pearson equation, which enables us to prove that the sequence of derivatives of the orthogonal polynomials is also orthogonal, and to compute a Rodrigues formula for these polynomials as well as a matrix-valued differential operator having these polynomials as eigenfunctions. We also describe the second-order differential operators of the algebra associated with the weight matrix.

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Elementary examples of solutions to Bochner’s problem for matrix differential operators

Cited by 4 publications

References 26 publications

Matrix-Valued Gegenbauer-Type Polynomials

Matrix-Valued Gegenbauer-Type Polynomials

Matrix Bispectrality and Noncommutative Algebras: beyond the prolate spheroidals

Structural Formulas for Matrix-Valued Orthogonal Polynomials Related to $$2\times 2$$ Hypergeometric Operators

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