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

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 rela...

Classes of Bivariate Orthogonal Polynomials

We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-D Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the 2-D Hermite polynomials and a two variable extension of the Zernike or disc polynomials. We also give q-analogues of all these extensions. In each case in addit...

Orthogonal polynomials and the finite Toda lattice

The choice of a finitely supported distribution is viewed as a degenerate bilinear form on the polynomials in the spectral parameter z and the matrix representing multiplication by z in terms of an orthogonal basis is constructed. It is then shown that the same induced time dependence for finitely supported distributions which gives the ith KP flow under the dual isomorphism induces the ith flo...

On Lax pairs and matrix extended simple Toda systems

Noncommutative theories have been studied and probed from different viewpoints (see reviews [18, 34, 48]). For instance, a number of noncommutative generalizations of integrable systems were presented (see, e.g., [9, 16, 17, 24, 39]). Solutions were investigated using the dressing method and Riemann-Hilbert problems, formulations, and properties such as infinite sets of conserved quantities wer...

Equations of motion for zeros of orthogonal polynomials related to the Toda lattices