Matrix biorthogonal polynomials: Eigenvalue problems and non-Abelian discrete Painleve equations A Riemann-Hilbert problem perspective



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Branquinho, Amilcar and Foulquié Moreno, Ana and Mañas Baena, Manuel (2021) Matrix biorthogonal polynomials: Eigenvalue problems and non-Abelian discrete Painleve equations A Riemann-Hilbert problem perspective. Journal of mathematical analysis and applications, 494 (2). ISSN 0022-247X

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In this paper we use the Riemann-Hilbert problem, with jumps supported on appropriate curves in the complex plane, for matrix biorthogonal polynomials and apply it to find Sylvester systems of differential equations for the orthogonal polynomials and its second kind functions as well. For this aim, Sylvester type differential Pearson equations for the matrix of weights are shown to be instrumental. Several applications are given, in order of increasing complexity. First, a general discussion of non-Abelian Hermite biorthogonal polynomials on the real line, understood as those whose matrix of weights is a solution of a Sylvester type Pearson equation with coefficients first degree matrix polynomials, is given. All of these are applied to the discussion of possible scenarios leading to eigenvalue problems for second order linear differential operators with matrix eigenvalues. Nonlinear matrix difference equations are discussed next. Firstly, for the general Hermite situation a general non linear relation (non trivial because of the non commutativity features of the setting) for the recursion coefficients is gotten. In the next case of higher difficulty, degree two polynomials are allowed in the Pearson equation, but the discussion is simplified by considering only a left Pearson equation. In the case, the support of the measure is on an appropriate branch of a hyperbola. The recursion coefficients are shown to fulfill a non-Abelian extension of the alternate discrete PainleveI equation. Finally, a discussion is given for the case of degree three polynomials as coefficients in the left Pearson equation characterizing the matrix of weights. However, for simplicity only odd polynomials are allowed. In this case, a new and more general matrix extension of the discrete Painleve I equation is found. (c) 2020 Elsevier Inc. All rights reserved.

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© 2021 Academic Press Inc Elsevier Science.
Acknowledges Centre for Mathematics of the University of Coimbra (Portuguese Government through FCT/MCTES) within project UIDB/00324/2020.; Acknowledges Center for Research and Development in Mathematics and Applications from University of Aveiro (Portuguese Government through FCT/MCTES) within project UIDB/04106/2020.; Acknowledges economical support from the Spanish Ministerio de Economia y Competitividadresearch project [MTM201565888-C4-2-P], Ortogonalidad, teoria de la aproximacion y aplicaciones en fisica matematicaand Spanish Agencia Estatal de Investigacionresearch project [PGC2018-096504-B-C33], Ortogonalidad y Aproximacion: Teoria y Aplicaciones en Fisica Matematica.

Uncontrolled Keywords:Valued orthogonal polynomials; Differential equations; Recurrence coefficients; Laurent polynomials; Unit circle; Asymptotics; Models; Formulas; Extensions; Respect.
Subjects:Sciences > Physics > Physics-Mathematical models
Sciences > Physics > Mathematical physics
ID Code:63254
Deposited On:02 Dec 2020 17:09
Last Modified:02 Dec 2020 17:09

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