Semiclassical orthogonal polynomials in two variables

Morales, María Álvarez de; Fernández, Lidia; Pérez, Teresa E.; Piñar, Miguel A.

doi:10.1016/j.cam.2006.10.017

Cited by 8 publications

(15 citation statements)

References 9 publications

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“…A study of two-variable orthogonal polynomials associated with a moment functional satisfying the two-variable analogue of the Pearson differential equation and an extension of some of the usual characterizations of the classical orthogonal polynomials in one variable was found [45]. In [8] semiclassical orthogonal polynomials in two variables are defined as the orthogonal polynomials associated with a quasi definite linear functional satisfying a matrix Pearson-type differential equation, semiclassical functionals are characterized by means of the analogue of the structure relation in one variable and non trivial examples of semiclassical orthogonal polynomials in two variables where given. Xu and Ilieva gave in [65] a characterization of all second order difference operators of several variables that have discrete orthogonal polynomials as eigenfunctions is given and under some mild assumptions, they give a complete solution of the problem.…”

Section: Introductionmentioning

confidence: 99%

Multivariate orthogonal polynomials and integrable systems

Ariznabarreta

Mañas

2016

Advances in Mathematics

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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 Christoffel-Darboux formulae. The multivariate orthogonal polynomials, its second kind functions and the corresponding Christoffel-Darboux kernels are shown to be quasi-determinants -as well as Schur complements-of bordered truncations of the moment matrix; quasi-tau 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 Zakharov-Shabat 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 Gauss-Borel factorization of the Jacobi type matrices and its quasideterminants, lead to expressions for the multivariate orthogonal polynomials and its second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm those that relate the Baker and adjoint Baker functions with ratios of Miwa shifted τ -functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants 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 quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involves only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials are 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.

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Section: Introductionmentioning

confidence: 99%

Multivariate orthogonal polynomials and integrable systems

Ariznabarreta

Mañas

2016

Advances in Mathematics

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“…The aim of our contribution is to analyse the extension of characterizations (a)-(c) to the multivariate case by means of a matrix formalism. In the bivariate case, the first of these three characterizations was the main result of a previous publication (see [1]). The two other characterizations constitute the main objective of this work.…”

Section: Introductionmentioning

confidence: 85%

“…Let us recall Theorem 5 in [1], which we need in the sequence of the paper, and constitutes a characterization for multivariate semiclassical orthogonal polynomials.…”

Section: Definitionmentioning

confidence: 99%

“…For a complete description of this and other related subjects see, for instance, [1,3,5,11,12,16,17].…”

Section: Introductionmentioning

confidence: 99%

“…Then u is called the moment functional determined by { } ∈N d 0 , and the number is called the moment of order . Let (P n ) | |=n be a sequence of polynomials of total degree n. Using the matrix notation introduced in [11,12,17], we can denote by P n the vector polynomial P n = (P n ) | |=n = (P n (1) , P n (2) , . .…”

Section: Introductionmentioning

confidence: 99%

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A semiclassical perspective on multivariate orthogonal polynomials

Morales

Fernández

Pérez

et al. 2008

Journal of Computational and Applied Mathematics

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Matrix Pearson Equations Satisfied by Koornwinder Weights in Two Variables

et al. 2017

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We consider Koornwinder's method for constructing orthogonal polynomials in two variables from orthogonal polynomials in one variable. If semiclassical orthogonal polynomials in one variable are used, then Koornwinder's construction generates semiclassical orthogonal polynomials in two variables. We consider two methods for deducing matrix Pearson equations for weight functions associated with these polynomials, and consequently, we deduce the second order linear partial differential operators for classical Koornwinder polynomials.

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Semiclassical orthogonal polynomials in two variables

Cited by 8 publications

References 9 publications

Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials and integrable systems

A semiclassical perspective on multivariate orthogonal polynomials

Matrix Pearson Equations Satisfied by Koornwinder Weights in Two Variables

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