Structure relations for monic orthogonal polynomials in two discrete variables

Rodal, J. J. A.; Area, Iván; Godoy, E.

doi:10.1016/j.jmaa.2007.09.003

Cited by 17 publications

(11 citation statements)

References 22 publications

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“…Recursive threeterm recurrence for the multivariate Jacobi polynomials on a simplex are explicitly given in [113]. In [97] several relations linking differences of bivariate discrete orthogonal polynomials and polynomials are given. We should also mention the work [36] in where bivariate real valued polynomials orthogonal with respect to a positive linear functional are considered; interestingly the authors discuss orthogonal polynomials associated with positive definite block Hankel matrices whose entries are also Hankel and develop methods for constructing such matrices.…”

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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“…T , are matrices of the size (2n + 2) × (n + 1) and (2n + 2) × n, respectively, which can be obtained by using (6.19) in terms of the coefficients of the partial q-difference equation (3.3), explicitly given in (3.4). This means that the recurrence (6.23) gives another realisation of [13, (3.2.10)], already appeared in the bivariate discrete case in [45]. Therefore, from (6.23) it is possible to compute a monic orthogonal polynomial solution of an admissible potentially self-adjoint linear second-order partial q-difference equation of the hypergeometric type (3.3).…”

Section: Monic Orthogonal Polynomial Solutionsmentioning

confidence: 97%

“…In more recent papers the second-order linear partial differential equations of the hypergeometric type [5] and their discretization on uniform lattices [4,6,44,45], as well as a general way of introducing orthogonal polynomial families in two discrete variables on the simplex [43], have been analyzed. Therefore, it is possible to generalize the univariate classical orthogonal polynomials to the bivariate and multivariate versions by requiring that they obey a second-order partial differential equation of the hypergeometric type (continuous case) [32,48], or a second-order partial difference equation of the hypergeometric type (discrete case), as indicated before.…”

Section: Introductionmentioning

confidence: 99%

Linear partial q-difference equations on q-linear lattices and their bivariate q-orthogonal polynomial solutions

Area

Atakishiyev

Godoy

et al. 2013

Applied Mathematics and Computation

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Orthogonal polynomial solutions of an admissible potentially self-adjoint linear second-order partial q-difference equation of the hypergeometric type in two variables on q-linear lattices are analyzed. A q-Pearson's system for the orthogonality weight function, as well as for the difference derivatives of the solutions are presented, giving rise to a solution of the q-difference equation under study in terms of a Rodriguestype formula. The monic orthogonal polynomial solutions are treated in detail, giving explicit formulae for the matrices in the corresponding recurrence relations they satisfy. Lewanowicz and Woźny [S. Lewanowicz, P. Woźny, J. Comput. Appl. Math. 233 (2010) 1554-1561] have recently introduced a (non-monic) bivariate extension of big q-Jacobi polynomials together with a partial q-difference equation of the hypergeometric type that governs them. This equation is analyzed in the last section: we provide two more orthogonal polynomial solutions, namely, a second non-monic solution from the Rodrigues' representation, and the monic solution both from the recurrence relation that govern them and also explicitly given in terms of generalized bivariate basic hypergeometric series. Limit relations as q ↑ 1 for the partial q-difference equation and for the all three q-orthogonal polynomial solutions are also presented.

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“…Next, a systematic study of the orthogonal polynomial solutions to a second order partial differential equation with two variables of hypergeometric type was made in [37]. In the bivariate discrete case, a hypergeometric formula was also given in [38].…”

Section: Related Workmentioning

confidence: 99%

A General Method for Generating Discrete Orthogonal Matrices

Chan

2021

IEEE Access

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Discrete orthogonal matrices have applications in information coding and cryptography. It is often challenging to generate discrete orthogonal matrices. A common approach widely in use is to discretize continuous orthogonal functions that have been discovered. The need of such continuous functions is restrictive. Polynomials, as the simplest class of continuous functions, are widely studied for their orthogonalality, to serve the purpose of generating orthogonal matrices. However, beginning with continuous orthogonal polynomials still takes much work. To overcome this complexity while improving the efficiency and flexibility, we present a general method for generating orthogonal matrices directly through the construction of certain even and odd polynomials from a set of distinct positive values, bypassing the need of continuous orthogonal functions. We present a constructive proof by induction that not only asserts the existence of such polynomials, but also tells how to iteratively construct them. Besides the derivation of the method as simple as a few nested loops, we discuss two well-known discrete transforms, the Discrete Cosine Transform and the Discrete Tchebichef Transform, how they can be achieved using our method with the specific values, and show how to embed them into the transform module of video coding. By the same token, we also give the examples for generating new orthogonal matrices from arbitrarily chosen values. The demonstrative experiments indicate that our method is not only simpler to implement, but also more efficient and flexible. It can generate orthogonal matrices of larger sizes, compared with those existing methods.

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Structure relations for monic orthogonal polynomials in two discrete variables

Cited by 17 publications

References 22 publications

Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials and integrable systems

Linear partial q-difference equations on q-linear lattices and their bivariate q-orthogonal polynomial solutions

A General Method for Generating Discrete Orthogonal Matrices

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