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Ariznabarreta, Gerardo and Mañas Baena, Manuel (2016) Multivariate orthogonal polynomials and integrable systems. Advances in mathematics, 302 . pp. 628739. ISSN 00018708

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Official URL: http://dx.doi.org/10.1016/j.aim.2016.06.029
URL  URL Type 

https://arxiv.org/abs/1409.0570  Organisation 
Abstract
Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also ChristoffelDarboux formulae. The multivariate orthogonal polynomials, their second kind functions and the corresponding ChristoffelDarboux kernels are shown to be quasideterminants as well as Schur complements of bordered truncations of the moment matrix; quasitau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and ZakharovShabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a GaussBorel factorization of the Jacobi type matrices and its quasideterminants, lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasitau matrices, which generalize to the multidimensional realm, those that relate the Baker and adjoint Baker functions to ratios of Miwa shifted taufunctions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasideterminants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in R^D, of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasideterminant. It is shown, using congruences in the space of semiinfinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial differencedifferential equations that involve only one site in the integrable lattice behaving as a KadomstevPetviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry.
Item Type:  Article 

Additional Information:  © Elsevier 2016. 
Uncontrolled Keywords:  Multivariate orthogonal polynomials; BorelGauss factorization; Quasideterminants; ChristoffelDarboux kernels; Darboux transformations; Christoffel formula; Quasitau matrices; Kernel polynomials; Integrable hierarchies; Toda equations; KP equations 
Subjects:  Sciences > Physics > Mathematical physics 
ID Code:  40377 
Deposited On:  25 Nov 2016 15:35 
Last Modified:  10 Dec 2018 15:09 
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