Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy
| dc.contributor.author | Álvarez Fernández, Carlos | |
| dc.contributor.author | Fidalgo Prieto, Ulises | |
| dc.contributor.author | Mañas Baena, Manuel Enrique | |
| dc.date.accessioned | 2023-06-20T03:56:21Z | |
| dc.date.available | 2023-06-20T03:56:21Z | |
| dc.date.issued | 2011-07-10 | |
| dc.description | ©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. | |
| dc.description.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. | |
| dc.description.department | Depto. de Física Teórica | |
| dc.description.faculty | Fac. de Ciencias Físicas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | Ministerio de Ciencia e Innovacion, Spain | |
| dc.description.sponsorship | Fundacao para a Ciencia e a Tecnologia of Portugal | |
| dc.description.sponsorship | Acciones Integradas Portugal | |
| dc.description.sponsorship | Ministerio de Ciencia e Innovacion de España | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/31481 | |
| dc.identifier.doi | 10.1016/j.aim.2011.03.008 | |
| dc.identifier.issn | 0001-8708 | |
| dc.identifier.officialurl | http://dx.doi.org/10.1016/j.aim.2011.03.008 | |
| dc.identifier.relatedurl | http://www.sciencedirect.com | |
| dc.identifier.relatedurl | http://arxiv.org/abs/1004.3916 | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/44697 | |
| dc.issue.number | 4 | |
| dc.journal.title | Advances in mathematics | |
| dc.language.iso | eng | |
| dc.page.final | 1525 | |
| dc.page.initial | 1451 | |
| dc.publisher | Elsevier | |
| dc.relation.projectID | FIS2008-00200 | |
| dc.relation.projectID | PT2009-0031 | |
| dc.relation.projectID | MTM2009-12740-C03-01 | |
| dc.relation.projectID | SFRH/BPD/62947/2009 | |
| dc.rights.accessRights | open access | |
| dc.subject.cdu | 51-73 | |
| dc.subject.keyword | Kadomtsev-petviashvili equation | |
| dc.subject.keyword | Discrete kp-hierarchy | |
| dc.subject.keyword | Random matrices | |
| dc.subject.ucm | Física-Modelos matemáticos | |
| dc.subject.ucm | Física matemática | |
| dc.title | Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy | |
| dc.type | journal article | |
| dc.volume.number | 227 | |
| dcterms.references | [1] M. Adler and P. van Moerbeke, Group factorization, moment matrices and Toda lattices, Int. Math. Res. Notices 12 (1997) 556-572. [2] M. Adler and P. van Moerbeke, Generalized orthogonal polynomials, discrete KP and Riemann–Hilbert problems, Commun. Math. Phys. 207 (1999) 589-620. [3] M. Adler and P. van Moerbeke, Vertex operator solutions to the discrete KP hierarchy, Commun. Math. Phys. 203 (1999) 185-210. [4] M. Adler and P. van Moerbeke, The spectrum of coupled random matrices, Ann. Math. 149 (1999) 921-976. [5] M. Adler and P. van Moerbeke, Darboux transforms on band matrices, weights and associated polynomials, Int. Math. Res. Not. 18 (2001) 935-984. [6] M. Adler, P. van Moerbeke and P. Vanhaecke, Moment matrices and multi-component KP, with applications to random matrix theory, Commun. Math. Phys. 286 (2009) 1-38. [7] C. Álvarez-Fernández, U. Fidalgo and M. Mañas, The multicomponent 2D Toda hierarchy: generalized matrix orthogonal polynomials, multiple orthogonal polynomials and Riemann–Hilbert problems. Inv. Prob. 26 (2010) 055009 (17 pp). [8] R. Apery. Irrationalite de ζ(2) et ζ(3), Astèrisque 61 (1979) 11-13. [9] M. J. Bergvelt and A. P. E. ten Kroode, Partitions, vertex operators constructions and multi-component KP equations, Pacific J. Math. 171 (1995) 23-88. [10] F. Beukers, Padé approximation in number theory, Lec. Not. Math. 888, Springer Verlag, Berlin, 1981, 90-99. [11] P.M. Bleher and A.B.J. Kuijlaars, Random matrices with external source and multiple orthogonal polynomials, Int. Math. Res. Not. 2004 (2004), 109-129. [12] G. Carlet, The extended bigraded Toda hierarchy J. Phys. A: Math. Gen. 39 (2006) 9411-9435. [13] J. Coates, On the algebraic approximation of functions, I, II, III. Indag. Math. 28 (1966) 421-461. [14] E. Daems and A. B. J. Kuijlaars, Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions, J. of Approx. Theory 146 (2007) 91-114. [15] E. Daems and A. B. J. Kuijlaars, A Christoffel–Darboux formula for multiple orthogonal polynomials, J. of Approx. Theory 130 (2004) 188-200. [16] E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, Operator approach to the Kadomtsev-Petviashvili equation. Transformation groups for soliton equations. III J. Phys. Soc. Jpn. 50, (1981) 3806-3812. [17] E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, Transformation groups for soliton equations. Euclidean Lie algebras and reduction of the KP hierarchy, Publ. Res. Inst. Math. Sci. 18 (1982) 1077-1110. [18] E. Date, M. Jimbo, M. Kashiwara, and T. Miwa, Transformation groups for soliton equations in Nonlinear Integrable Systems-Classical Theory and Quantum Theory M. Jimbo and T. Miwa (eds.) World Scientific, Singapore, 1983, pp. 39-120. [19] R. Felipe F. Ongay, Algebraic aspects of the discrete KP hierarchy, Linear Algebra App. 338 (2001) 1-17. [20] U. Fidalgo Prieto and G. López Lagomasino, Nikishin systems are perfect, arXiv:1001.0554 (2010). [21] A. S. Fokas, A. R. Its and A. V. Kitaev, The isomonodromy approach to matrix models in 2D quatum gravity, Commun. Math. Phys. (1992) 395-430. [22] Ch. Hermite, Sur la fonction exponentielle, C. R. Acad. Sci. Paris 77 (1873), 18-24, 74-79, 226-233, 285-293; reprinted in his Oeuvres, Tome III, Gauthier-Villars, Paris, 1912, 150-181. [23] H. Jager, A simultaneous generalization of the Padé table, I- VI, Indag. Math. 26 (1964), 193-249. [24] V. G. Kac and J. W. van de Leur, The n-component KP hierarchy and representation theory, J. Math. Phys. 44 (2003) 3245-3293. [25] A. B. J. Kuijlaars, Multiple orthogonal polynomial ensembles, arXiv:0902.1058v1 [math.PR] (2009). [26] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1995. [27] K. Mahler, Perfect systems, Compos. Math. 19 (1968), 95- 166. [28] M. Mañas, L. Martínez Alonso and C. Alvarez-Fernández, The multicomponent 2D Toda hierarchy: discrete flows and string equations, Inv. Prob. 25 (2009) 065007 (31 pp). [29] M. Mañas and L. Martínez Alonso, The multicomponent 2D Toda hierarchy: dispersionless limit, Inv. Prob. 25 (2009) 115020 (22 pp). [30] M. Mulase, Complete integrability of the Kadomtsev– Petviashvili equation, Adv. Math. 54 (1984) 57-66. [31] E. M. Nikishin, On simultaneous Padé approximants Matem. Sb. 113 (1980), 499–519 (Russian); English translation in Math. USSR Sb. 41 (1982), 409-425. [32] A. Yu. Orlov and D. M. Scherbin, Fermionic representation for basic hypergeometric functions related to Schur polynomials, arXiv:nlin/0001001v4 [nlin.SI]. [33] M. Sato, Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds, Res. Inst. Math. Sci. Kokyuroku 439, (1981) 30-46 . [34] J. A. Shohat and J.D. Tamarkin, The problem of moments, American Mathematical Society (1943). [35] B. Simon, The Christoffel-Darboux Kernel, Proceedings of Symposia in Pure Mathematics 79:“Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G. Maz’ya’s 70th Birthday”, (2008) 295-336. arXiv:0806.1528 [36] V.N. Sorokin, On simultaneous approximation of several linear forms, Vestnik Mosk Univ., Ser Matem 1 (1983) 44-47. [37] K. Ueno and K. Takasaki, Toda lattice hierarchy, in Group Representations and Systems of Differential Equations, Adv. Stud. Pure Math. 4 (1984) 1-95. [38] R. P. Stanley, Enumerative combinatorics, Cambridge University Press, Cambridge (1998). [39] W. Van Assche, J. S. Geromino and A. B. J. Kuijlaars, Riemann–Hilbert problems for multiple orthogonal polynomials in: Bustoz et al (eds.), Special Functions 2000: Current Perspectives and Future Directions, Kluwer Academic Publishers, Dordrecht, 2001, pp 23-59. | |
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