Weak classical orthogonal polynomials in two variables

Fernández, Lidia; Pérez, Teresa E.; Piñar, Miguel A.

doi:10.1016/j.cam.2004.08.010

Cited by 20 publications

(24 citation statements)

References 14 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.…”

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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“…These operators can be extended for higher order derivatives [1], and for matrices [5,6]. In fact, if we denote…”

Section: Definition 32 a Weak Orthogonal Polynomial System (Wops) Asmentioning

confidence: 99%

“…Recently, the authors (see [5,6]) extended the concept of classical orthogonal polynomials in two variables to a wider framework, which, of course, includes the Krall and Sheffer definition and tensor products of classical orthogonal polynomials in one variable. Our approach to classical orthogonal polynomials in two variables starts from a quasi-definite moment functional u satisfying a matrix Pearson-type differential equation…”

Section: Introductionmentioning

confidence: 99%

On differential properties for bivariate orthogonal polynomials

et al. 2007

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“…As indicated above denotẽ 5) where the (n + 1) × [(n + 1)(m + 1)] matrixK n,m is given similarly to (3.4) with the roles of n and m interchanged. In order to find recurrence formulas for the vector polynomials P n,m we introduce the inner product,…”

Section: Lexicographic Order and Orthogonal Polynomialsmentioning

confidence: 99%

“…Special examples of these types of polynomials have arisen in studies related to symmetry groups (Dunkl [3], Koornwinder [14], MacDonald [20]), as extensions of one variable polynomials (Fernández-Pérez-Piñar [5], Koornwinder [13]) and as eigenfunctions of partial differential equations (Koornwinder [12], Krall-Sheffer [17], Kim-Kwon-Lee [11], Kwon-Lee-Littlejohn [19] (see also the references in [4])). The general theory of these polynomials can trace its origins back to Jackson [10] and an excellent review of the theory can be found in the book of Dunkl and Xu [4] (see also the book of Suetin [21]).…”

Section: Introductionmentioning

confidence: 99%