Linear spectral transformations for multivariate orthogonal polynomials and multispectral Toda hierarchies

TL;DR

This paper extends linear spectral transformations to multivariate orthogonal polynomials, deriving new formulas and introducing a multispectral Toda hierarchy linked to multivariate biorthogonality and integrable systems.

Contribution

It develops multivariate Christoffel-Geronimus-Uvarov formulas and introduces a novel multispectral Toda hierarchy for multivariate orthogonal polynomials.

Findings

Derived multivariate Christoffel-Geronimus-Uvarov formulas.

Formulated a new multispectral Toda hierarchy.

Connected multivariate orthogonal polynomials with integrable systems.

Abstract

Linear spectral transformations of orthogonal polynomials in the real line, and in particular Geronimus transformations, are extended to orthogonal polynomials depending on several real variables. Multivariate Christoffel-Geronimus-Uvarov formulae for the perturbed orthogonal polynomials and their quasi-tau matrices are found for each perturbation of the original linear functional. These expressions are given in terms of quasi-determinants of bordered truncated block matrices and the 1D Christoffel-Geronimus-Uvarov formulae in terms of quotient of determinants of combinations of the original orthogonal polynomials and their Cauchy transforms, are recovered. A new multispectral Toda hierarchy of nonlinear partial differential equations, for which the multivariate orthogonal polynomials are reductions, is proposed. This new integrable hierachy is associated with non-standard multivariate biorthogonality. Wave and Baker functions, linear equations, Lax and Zakharov-Shabat equations, KP type equations, appropriate reductions, Darboux/linear spectral transformations, and bilinear equations involving linear spectral transformations are presented.