Bivariate Orthogonal Polynomials Research Articles

Centrosymmetric and reverse matrices in bivariate orthogonal polynomials

We study bivariate orthogonal polynomials associated with an inner product that satisfies a symmetry property such that it is invariant when both variables are interchanged. Under that hypothesis, the structure of the polynomial vectors of the orthogonal polynomial systems is described by using centrosymmetric matrices. Also, we prove that the coefficient matrices of the three-term relations, one for each variable, are connected by a reverse operation over matrices. Finally, several particular cases and examples are analysed.

Lax-type pairs in the theory of bivariate orthogonal polynomials

Sequences of bivariate orthogonal polynomials written as vector polynomials of increasing size satisfy a couple of three term relations with matrix coefficients. In this work, introducing a time-dependent parameter, we analyze a Lax-type pair system for the coefficients of the three term relations. We also deduce several characterizations relating the Lax-type pair, the shape of the weight, Stieltjes function, moments, a differential equation for the weight, and the bidimensional Toda-type systems.

Extended homogeneous bivariate orthogonal polynomials: symbolic and numerical Gaussian cubature formula

Extended homogeneous bivariate orthogonal polynomials: symbolic and numerical Gaussian cubature formula

Backtesting Expected Shortfall: Accounting for Both Duration and Severity with Bivariate Orthogonal Polynomials

Backtesting Expected Shortfall: Accounting for Both Duration and Severity with Bivariate Orthogonal Polynomials

A bivariate Q-polynomial structure for the non-binary Johnson scheme

The notion of multivariate P- and Q-polynomial association scheme has been introduced recently, generalizing the well-known univariate case. Numerous examples of such association schemes have already been exhibited. In particular, it has been demonstrated that the non-binary Johnson scheme is a bivariate P-polynomial association scheme. We show here that it is also a bivariate Q-polynomial association scheme for some parameters. This provides, with the P-polynomial structure, the bispectral property (i.e. the recurrence and difference relations) of a family of bivariate orthogonal polynomials made out of univariate Krawtchouk and dual Hahn polynomials. The algebra based on the bispectral operators is also studied together with the subconstituent algebra of this association scheme.

Orthogonal polynomials on a class of planar algebraic curves

AbstractWe construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form in where and ϕ is a polynomial of arbitrary degree d, in terms of univariate semiclassical OPs. We compute connection coefficients that relate the bivariate OPs to a polynomial basis that is itself orthogonal and whose span contains the OPs as a subspace. The connection matrix is shown to be banded and the connection coefficients and Jacobi matrices for OPs of degree are computed via the Lanczos algorithm in operations.

Quadratic Decomposition of Bivariate Orthogonal Polynomials

We describe the relation between the systems of bivariate orthogonal polynomial associated to a symmetric weight function and associated to some particular Christoffel modifications of the quadratic decomposition of the original weight. We analyze the construction of a symmetric bivariate orthogonal polynomial sequence from a given one, orthogonal to a weight function defined on the first quadrant of the plane. In this description, a sort of Bäcklund type matrix transformations for the involved three term matrix coefficients plays an important role. Finally, we take as a case study relations between the classical orthogonal polynomials defined on the ball and those on the simplex.

Fourier Transform of Orthogonal Polynomials over the Triangle with Four Parameters

In this paper, some new families of orthogonal functions in two variables produced by using Fourier transform of bivariate orthogonal polynomials and their orthogonality relations obtained from Parseval identity are introduced.

Two variable Freud orthogonal polynomials and matrix Painlevé-type difference equations

We study bivariate orthogonal polynomials associated with Freud weight functions depending on real parameters. We analyse relations between the matrix coefficients of the three term relations for the orthonormal polynomials as well as the coefficients of the structure relations satisfied by these bivariate semiclassical orthogonal polynomials, also a matrix differential-difference equation for the bivariate orthogonal polynomials is deduced. The extension of the Painlevé equation for the coefficients of the three term relations to the bivariate case and a two dimensional version of the Langmuir lattice are obtained.

Monic Bivariate Polynomials on Quadratic and q-Quadratic Lattices

Monic families of bivariate Askey–Wilson polynomials and q-Racah polynomials are explicitly given. Monic families of bivariate orthogonal polynomials, both in quadratic and q-quadratic lattices, are also explicitly given using appropriate limit relations.

Bivariate Koornwinder\u2013Sobolev Orthogonal Polynomials

The so-called Koornwinder bivariate orthogonal polynomials are generated by means of a non-trivial procedure involving two families of univariate orthogonal polynomials and a function rho (t) such that rho (t)^2 is a polynomial of degree less than or equal to 2. In this paper, we extend the Koornwinder method to the case when one of the univariate families is orthogonal with respect to a Sobolev inner product. Therefore, we study the new Sobolev bivariate families obtaining relations between the classical original Koornwinder polynomials and the Sobolev one, deducing recursive methods in order to compute the coefficients. The case when one of the univariate families is classical is analysed. Finally, some useful examples are given.

Fourier Transforms of Some Finite Bivariate Orthogonal Polynomials

In this paper, we first obtain the Fourier transforms of some finite bivariate orthogonal polynomials and then by using the Parseval identity, we introduce some new families of bivariate orthogonal functions.

Quasi-birth-and-death processes and multivariate orthogonal polynomials

The aim of this paper is to study some models of quasi-birth-and-death (QBD) processes arising from the theory of bivariate orthogonal polynomials. First we will see how to perform the spectral analysis in the general setting as well as to obtain results about recurrence and the invariant measure of these processes in terms of the spectral measure supported on some domain Ω⊂Rd. Afterwards, we will apply our results to several examples of bivariate orthogonal polynomials, namely product orthogonal polynomials, orthogonal polynomials on a parabolic domain and orthogonal polynomials on the triangle. We will focus on linear combinations of the Jacobi matrices generated by these polynomials and produce families of either continuous or discrete-time QBD processes. Finally, we show some urn models associated with these QBD processes.

A Sparse Spectral Method for Volterra Integral Equations Using Orthogonal Polynomials on the Triangle

We introduce and analyse a sparse spectral method for the solution of Volterra integral equations using bivariate orthogonal polynomials on a triangle domain. The sparsity of the Volterra operator on a weighted Jacobi basis is used to achieve high efficiency and exponential convergence. The discussion is followed by a demonstration of the method on example Volterra integral equations of the first and second kind with known analytic solutions as well as an application-oriented numerical experiment. We prove convergence for both first and second kind problems, where the former builds on connections with Toeplitz operators.

Geronimus transformations of bivariate linear functionals

Given a linear functional u in the linear space of polynomials in two variables with real coefficients and a polynomial λ(x,y), in this contribution we deal with Geronimus transformations of u, i.e., those linear functionals v such that u=λ(x,y)v. The connection formulas between the sequences of bivariate orthogonal polynomials with respect to u and v are given. A matrix interpretation of such transformations by using LU and UL factorizations of the block Jacobi matrices associated with such polynomials is given. Finally, some illustrative examples of Geronimus transformations of weight function supported in domains of R2 are discussed.

Coherent pairs of bivariate orthogonal polynomials

Coherent pairs of measures were introduced in 1991 and constitute a very useful tool in the study of Sobolev orthogonal polynomials on the real line. In this work, coherence and partial coherence in two variables appear as the natural extension of the univariate case. Given two families of bivariate orthogonal polynomials expressed as polynomial systems, they are a partial coherent pair if a polynomial of the second family can be given as a linear combination of the first partial derivatives of (at most) three consecutive polynomials of the first family. A full coherent pair is a pair of families of bivariate orthogonal polynomials related by means of partial coherent relations in each variable. Consequences of this kind of relations concerning both families of bivariate orthogonal polynomials are studied. Finally, some illustrative examples are provided.

A Sparse Spectral Method on Triangles

In this paper, we demonstrate that many of the computational tools for univariate orthogonal polynomials have analogues for a family of bivariate orthogonal polynomials on the triangle, including C...

On bivariate classical orthogonal polynomials

We deduce new characterizations of bivariate classical orthogonal polynomials associated with a quasi-definite moment functional, and we revise old properties for these polynomials. More precisely, new characterizations of classical bivariate orthogonal polynomials satisfying a diagonal Pearson-type equation are proved: they are solutions of two separate partial differential equations one for every partial derivative, their partial derivatives are again orthogonal, and every vector polynomial can be expressed in terms of its partial derivatives by means of a linear relation involving only three terms of consecutive total degree. Moreover, we study general solutions of the matrix second order partial differential equation satisfied by classical orthogonal polynomials, and we deduce the explicit expressions for the matrix coefficients of the structure relation. Finally, some illustrative examples are given.

Bivariate orthogonal polynomials, 2D Toda lattices and Lax-type pairs

We explore the connection between an infinite system of particles in R2 described by a bi-dimensional version of the Toda equations with the theory of orthogonal polynomials in two variables. We define a 2D Toda lattice in the sense that we consider only one time variable and two space variables describing a mesh of interacting particles over the plane. We show that this 2D Toda lattice is related with the matrix coefficients of the three term relations of bivariate orthogonal polynomials associated with an exponential modification of a positive measure. Moreover, block Lax pairs for 2D Toda lattices are deduced.

Three Term Relations for a Class of Bivariate Orthogonal Polynomials

We study matrix three term relations for orthogonal polynomials in two variables constructed from orthogonal polynomials in one variable. Using the three term recurrence relation for the involved univariate orthogonal polynomials, the explicit expression for the matrix coefficients in these three term relations are deduced. These matrices are diagonal or tridiagonal with entries computable from the one variable coefficients in the respective three term recurrence relation. Moreover, some interesting particular cases are considered.