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P. Túrán (1975)
On orthogonal polynomialsAnalysis Mathematica, 1
F. Grünbaurn, I. Pacharoni, J. Tirao (2003)
Matrix valued orthogonal polynomials of the Jacobi typeIndagationes Mathematicae, 14
F. Grünbaum, I. Pacharoni, I. Zurrián (2014)
Time and band limiting for matrix valued functions, an exampleSymmetry Integrability and Geometry-methods and Applications, 11
M. Castro, F. Grunbaum, I. Pacharoni, I. Zurri'an (2017)
A further look at time-and-band limiting for matrix orthogonal polynomialsarXiv: Classical Analysis and ODEs
F. Grünbaum, Manuel Iglesia (2008)
Matrix Valued Orthogonal Polynomials Arising from Group Representation Theory and a Family of Quasi-Birth-and-Death ProcessesSIAM J. Matrix Anal. Appl., 30
(1971)
Fundamental aspects of the representation theory of hermitian operators with deficiency index (m,m)
Antonio Guardeño, P. Rodríguez (2004)
Orthogonal matrix polynomials
M. Cantero, L. Moral, L. Velázquez (2006)
Matrix orthogonal polynomials whose derivatives are also orthogonalJ. Approx. Theory, 146
Antonio Guardeño, M. Iglesia (2008)
Some examples of orthogonal matrix polynomials satisfying odd order differential equationsJ. Approx. Theory, 150
I. Pacharoni, J. Tirao (2007)
Matrix Valued Orthogonal Polynomials Arising from the Complex Projective SpaceConstructive Approximation, 25
Á. Fantino, S. Grosz, D. Skigin (2004)
Shape resonances in nested diffraction gratingsOptik, 116
A. Durán (2009)
Generating orthogonal matrix polynomials satisfying second order differential equations from a trio of triangular matricesJournal of Approximation Theory, 161
J. Tirao (2003)
The matrix-valued hypergeometric equationProceedings of the National Academy of Sciences of the United States of America, 100
I. Pacharoni, I. Zurrián (2015)
Matrix Gegenbauer Polynomials: The 2 × 2 Fundamental Cases
(2019)
The symmetric 2 × 2 hypergeometric matrix differential operators
A. Durán (1997)
Matrix Inner Product Having a Matrix Symmetric Second Order Differential OperatorRocky Mountain Journal of Mathematics, 27
F. Grünbaum (2003)
Matrix valued Jacobi polynomialsBulletin Des Sciences Mathematiques, 127
F. Grunbaum, I. Pacharoni, I. Zurri'an (2018)
Bispectrality and Time–Band Limiting: Matrix-valued PolynomialsInternational Mathematics Research Notices
Jorge Borrego, M. Castro, Antonio Dur'an (2011)
Orthogonal matrix polynomials satisfying differential equations with recurrence coefficients having non-scalar limitsIntegral Transforms and Special Functions, 23
A. Durán (1996)
Markov's Theorem for Orthogonal Matrix PolynomialsCanadian Journal of Mathematics, 48
I. Pacharoni, P. Román (2007)
A Sequence of Matrix Valued Orthogonal Polynomials Associated to Spherical FunctionsConstructive Approximation, 28
E. Koelink, Maarten Pruijssen, P. Román (2010)
Matrix Valued Orthogonal Polynomials related to (SU(2) x SU(2),diag)Information Processing and Management
F. Grünbaum, I. Pacharoni, J. Tirao (2001)
Matrix Valued Spherical Functions Associated to the Complex Projective PlaneJournal of Functional Analysis, 188
M. Castro, F. Grünbaum (2006)
The algebra of differential operators associated to a family of matrix-valued orthogonal polynomials: Five instructive examplesInternational Mathematics Research Notices, 2006
Antonio Durán, F. Grünbaum (2005)
Structural Formulas for Orthogonal Matrix Polynomials Satisfying Second-Order Differential Equations, IConstructive Approximation, 22
A. Durán, F. Grünbaum (2007)
Matrix orthogonal polynomials satisfying second-order differential equations: Coping without help from group representation theoryJ. Approx. Theory, 148
I︠u︡. Berezanskiĭ, R. Bolstein (1968)
Expansions in eigenfunctions of selfadjoint operators
A. Durán (2010)
Rodrigues’ Formulas for Orthogonal Matrix Polynomials Satisfying Second-Order Differential EquationsInternational Mathematics Research Notices, 2010
W. Casper, M. Yakimov (2018)
The matrix Bochner problemAmerican Journal of Mathematics, 144
M. Krein (2016)
Infinite J-matrices and a matrix moment problemarXiv: Classical Analysis and ODEs
(2005)
On classical orthogonal polynomials and differential operatorsJournal of Physics A, 38
A. Durán, P. Lopez-Rodriguez (2007)
Structural Formulas for Orthogonal Matrix Polynomials Satisfying Second-Order Differential Equations, IIConstructive Approximation, 26
I. Zurri'an (2015)
The Algebra of Differential Operators for a Gegenbauer Weight MatrixarXiv: Classical Analysis and ODEs
I. Pacharoni, I. Zurrián (2013)
Matrix Gegenbauer Polynomials: The $$2\times 2$$2×2 Fundamental CasesConstructive Approximation, 43
J. Duistermaat, F. Grünbaum (1986)
Differential equations in the spectral parameterCommunications in Mathematical Physics, 103
(2008)
and M
MG Krein (1971)
Fundamental aspects of the representation theory of hermitian operators with deficiency index (m,m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(m, m)$$\end{document}AMS Transl. Ser., 2
M. Castro, F. Grünbaum (2015)
The Darboux process and time-and-band limiting for matrix orthogonal polynomials☆Linear Algebra and its Applications, 487
Celeste Calderón, Yanina González, I. Pacharoni, S. Simondi, I. Zurrián (2018)
2 × 2 Hypergeometric Operators with Diagonal EigenvaluesJ. Approx. Theory, 248
W. Casper (2015)
Elementary examples of solutions to Bochner's problem for matrix differential operatorsJ. Approx. Theory, 229
E. Koelink, A. Ríos, P. Román (2017)
Matrix-Valued Gegenbauer-Type PolynomialsConstructive Approximation, 46
M. Castro, F. Grünbaum (2017)
Time-and-band limiting for matrix orthogonal polynomials of Jacobi type, 06
J. Tirao, I. Zurrián (2015)
Reducibility of matrix weightsThe Ramanujan Journal, 45
F. Grünbaum, I. Pacharoni, J. Tirao (2001)
A matrix-valued solution to Bochner's problemJournal of Physics A: Mathematical and General, 34
F. Grünbaum, J. Tirao (2007)
The Algebra of Differential Operators Associated to a Weight MatrixIntegral Equations and Operator Theory, 58
FA Grünbaum, I Pacharoni, I Zurrián (2017)
Time and band limiting for matrix-valued functionsInverse Probl., 33
A. Durán (2009)
A Method to Find Weight Matrices Having Symmetric Second-Order Differential Operators with Matrix Leading CoefficientConstructive Approximation, 29
FA Grünbaum, I Pacharoni, I Zurrián (2015)
Time and band limiting for matrix-valued functions, an exampleSIGMA Symmetry Integr. Geom. Methods Appl., 11
(1964)
Discrete and Continous Boundary Problems
A. Durán, F. Grünbaum (2004)
Orthogonal matrix polynomials satisfying second-order differential equationsInternational Mathematics Research Notices, 2004
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We give some structural formulas for the family of matrix-valued orthogonal polynomials of size 2×2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$2\times 2$$\end{document} 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.
Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals
Published: Mar 1, 2022
Keywords: Matrix-valued orthogonal polynomials; Matrix-valued differential operators; Rodrigues formula; 42C05; 47S10; 33C45
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