Multivariate orthogonal polynomial and integrable systems

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 formulæ. 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 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 $\tau$-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 ${\mathbb{R}}^D$, of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of $m$ elementary Darboux transformations is given as a quasi-determinant.

In this third version we correct a typo

References in corpus (11)

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• Matrix Orthogonal Laurent Polynomials on the Unit Circle and Toda Type Integrable Systems
• Universal Whitham hierarchy, dispersionless Hirota equations and multi-component KP hierarchy
• Christoffel transformations for matrix orthogonal polynomials in the real line and the non-Abelian 2D Toda lattice hierarchy
• Darboux transformations for multivariate orthogonal polynomials
• On the Christoffel--Darboux formula for generalized matrix orthogonal polynomials of multigraded-Hankel type
• Singularity confinement for matrix discrete Painleve Equations
• Another Christoffel--Darboux Formula for Multiple Orthogonal Polynomials of Mixed Type
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Cited by in corpus (15)

• Orthogonal polynomials of several variables
• Christoffel transformations for matrix orthogonal polynomials in the real line and the non-Abelian 2D Toda lattice hierarchy
• Darboux transformations for multivariate orthogonal polynomials
• Transformation theory and Christoffel formulas for matrix biorthogonal polynomials on the real line
• Multivariate Orthogonal Polynomials and Modified Moment Functionals
• Isospectral flows related to Frobenius-Stickelberger-Thiele polynomials
• Multiple Orthogonal Polynomials and Random Walks
• Hypergeometric Multiple Orthogonal Polynomials and Random Walks
• Linear spectral transformations for multivariate orthogonal polynomials and multispectral Toda hierarchies
• Non-autonomous multidimensional Toda system and multiple interpolation problem
• The Toda and Painlevé Systems Associated with Semiclassical Matrix-Valued Orthogonal Polynomials of Laguerre Type
• CMV biorthogonal Laurent polynomials: Christoffel formulas for Christoffel and Geronimus perturbations
• CMV biorthogonal Laurent polynomials. II: Christoffel formulas for Geronimus-Uvarov perturbations
• Matrix Orthogonal Polynomials, non-abelian Toda lattice and Bäcklund transformation
• Christoffel transformations for (partial-)skew-orthogonal polynomials and applications