Christoffel Transformations for Matrix Orthogonal Polynomials in the Real Line and the non-Abelian 2D Toda Lattice Hierarchy

Álvarez-Fernández, Carlos; Ariznabarreta, Gerardo; García-Ardila, Juan C.; Mañas, Manuel; Marcellán, Francisco

doi:10.1093/imrn/rnw027

Cited by 45 publications

(75 citation statements)

References 96 publications

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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%

“…We actually do not need the explicit expression for Φ (ν) to prove Theorem 2.4, but we give it for completeness in (4.9) since it gives a highly nontrivial example of the theory for Christoffel transformations for matrix-valued weights as presented in [4].…”

Section: Pearson Equation For the Weight W (ν)mentioning

confidence: 99%

“…Matrix-valued orthogonal polynomials have applications in the study of higher-order recurrences and their spectral analysis (see, e.g., [2,16,19,20]) and the study of Toda lattices [4,6,18], to name a few.…”

Section: Introductionmentioning

confidence: 99%

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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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“…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%

Section: Pearson Equation For the Weight W (ν)mentioning

confidence: 99%

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Matrix-Valued Gegenbauer-Type Polynomials

2017

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“…Recently, in Ref. , we have extended the Christoffel transformation to MOPs in the real line (MOPRL) obtaining a new matrix Christoffel formula, and in Refs. and , more general transformations of Geronimus and Uvarov type are studied.…”

Section: Introductionmentioning

confidence: 99%

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

Cassatella‐Contra

Mañas

2019

Stud Appl Math

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Matrix Szegő biorthogonal polynomials for quasi‐definite matrices of Hölder continuous weights are studied. A Riemann‐Hilbert problem is uniquely solved in terms of the matrix Szegő polynomials and its Cauchy transforms. The Riemann‐Hilbert problem is given as an appropriate framework for the discussion of the Szegő matrix and the associated Szegő recursion relations for the matrix orthogonal polynomials and its Cauchy transforms. Pearson‐type differential systems characterizing the matrix of weights are studied. These are linear systems of ordinary differential equations that are required to have trivial monodromy. Linear ordinary differential equations for the matrix Szegő polynomials and its Cauchy transforms are derived. It is shown how these Pearson systems lead to nonlinear difference equations for the Verblunsky matrices and two examples, of Fuchsian and non‐Fuchsian type, are considered. For both cases, a new matrix version of the discrete Painlevé II equation for the Verblunsky matrices is found. Reductions of these matrix discrete Painlevé II systems presenting locality are discussed.

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“…(i) In [61,62], we consider some extensions of the Christoffel-Darboux formula to generalized orthogonal polynomials [51] and to multiple orthogonal polynomials, respectively. (ii) Matrix orthogonal polynomials, its Christoffel transformations, and the relation with non-Abelian Toda hierarchies were studied in [63], and in [64] we extended those results to include the Geronimus, Geronimus-Uvarov, and Uvarov transformations. (iii) Multiple orthogonal polynomials and multicomponent Toda [65].…”

Section: Introductionmentioning

confidence: 99%

CMV Biorthogonal Laurent Polynomials: Perturbations and Christoffel Formulas

Ariznabarreta¹,

Mañas²,

Toledano³

2018

Stud Appl Math

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Quasidefinite sesquilinear forms for Laurent polynomials in the complex plane and corresponding CMV biorthogonal Laurent polynomial families are studied. Bivariate linear functionals encompass large families of orthogonalities such as Sobolev and discrete Sobolev types. Two possible Christoffel transformations of these linear functionals are discussed. Either the linear functionals are multiplied by a Laurent polynomial, or are multiplied by the complex conjugate of a Laurent polynomial. For the Geronimus transformation, the linear functional is perturbed in two possible manners as well, by a division by a Laurent polynomial or by a complex conjugate of a Laurent polynomial, in both cases the addition of appropriate masses (linear functionals supported on the zeros of the perturbing Laurent polynomial) is considered. The connection formulas for the CMV biorthogonal Laurent polynomials, its norms, and Christoffel-Darboux kernels, in all the four cases, are given. For the Geronimus transformation, the connection formulas for the second kind functions and mixed Christoffel-Darboux kernels are also given in the two possible cases. For prepared Laurent polynomials, i.e., of the form L(z) = L n z n + · · · + L −n z −n , L n L −n = 0, these connection formulas lead to quasideterminantal (quotient of determinants) Christoffel formulas for all the four transformations, expressing an arbitrary degree perturbed biorthogonal Laurent polynomial in terms of 2n unperturbed biorthogonal Laurent polynomials, their second kind functions or Christoffel-Darboux kernels and its mixed versions. Different curves are presented as examples, such as the real line, the circle, the Cassini oval,

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Christoffel Transformations for Matrix Orthogonal Polynomials in the Real Line and the non-Abelian 2D Toda Lattice Hierarchy

Cited by 45 publications

References 96 publications

Matrix-Valued Gegenbauer-Type Polynomials

Matrix-Valued Gegenbauer-Type Polynomials

Riemann‐Hilbert problem and matrix discrete Painlevé II systems

CMV Biorthogonal Laurent Polynomials: Perturbations and Christoffel Formulas

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