# Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy

### Impacto

Álvarez Fernández, Carlos and Fidalgo Prieto, Ulises and Mañas Baena, Manuel (2011) Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy. Advances in mathematics, 227 (4). pp. 1451-1525. ISSN 0001-8708

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Official URL: http://dx.doi.org/10.1016/j.aim.2011.03.008

## Abstract

Multiple orthogonality is considered in the realm of a Gauss-Borel factorization problem for a semi-infinite moment matrix. Perfect combinations of weights and a finite Borel measure are constructed in terms of M-Nikishin systems. These perfect combinations ensure that the problem of mixed multiple orthogonality has a unique solution, that can be obtained from the solution of a Gauss-Borel factorization problem for a semi-infinite matrix, which plays the role of a moment matrix. This leads to sequences of multiple orthogonal polynomials, their duals and second kind functions. It also gives the corresponding linear forms that are bi-orthogonal to the dual linear forms. Expressions for these objects in terms of determinants from the moment matrix are given, recursion relations are found, which imply a multi-diagonal Jacobi type matrix with snake shape, and results like the ABC theorem or the Christoffel-Darboux formula are re-derived in this context (using the factorization problem and the generalized Hankel symmetry of the moment matrix). The connection between this description of multiple orthogonality and the multi-component 2D Toda hierarchy, which can be also understood and studied through a Gauss-Borel factorization problem, is discussed. Deformations of the weights, natural for M-Nikishin systems, are considered and the correspondence with solutions to the integrable hierarchy, represented as a collection of Lax equations, is explored. Corresponding Lax and Zakharov-Shabat matrices as well as wave functions and their adjoints are determined. The construction of discrete flows is discussed in terms of Miwa transformations which involve Darboux transformations for the multiple orthogonality conditions. The bilinear equations are derived and the tau-function representation of the multiple orthogonality is given.

Item Type: Article ©Alvarez-Fernandez2011 Elsevier Inc. All rights reserved.MM thanks economical support from the Spanish Ministerio de Ciencia e Innovación, research project FIS2008-00200 and UF thanks economical support from the SFRH / BPD / 62947 / 2009, Fundação para a Ciência e a Tecnologia of Portugal, PT2009-0031, Acciones Integradas Portugal, Ministerio de Ciencia e Innovación de España, and MTM2009- 12740-C03-01. Ministerio de Ciencia e Innovación de España. MM reckons different and clarifying discussions with Dr. Mattia Caffasso, Prof. Pierre van Moerbeke, Prof. Luis Martínez Alonso and Prof. David Gómez-Ullate. The authors of this paper are in debt with Prof. Guillermo López Lagomasino who carefully read the manuscript and whose suggestions improved the paper indeed. Kadomtsev-petviashvili equation; Discrete kp-hierarchy; Random matrices Sciences > Physics > Physics-Mathematical modelsSciences > Physics > Mathematical physics 31481 17 Jul 2015 12:19 10 Dec 2018 15:09