Christoffel transformations for matrix orthogonal polynomials in the real line and the non-Abelian 2D Toda lattice hierarchy

Abstract

Given a matrix polynomial $W(x)$, matrix bi-orthogonal polynomials with respect to the sesquilinear form $⟨P(x),Q(x){⟩}_W=\int P(x)W(x)d\mu (x)(Q(x){)}^{\top }$, $P(x),Q(x)\in {\mathbb{R}}^{p\times p}[x]$, where $\mu (x)$ is a matrix of Borel measures supported in some infinite subset of the real line, are considered. Connection formulas between the sequences of matrix bi-orthogonal polynomials with respect to $⟨\cdot ,\cdot {⟩}_W$ and matrix polynomials orthogonal with respect to $\mu (x)$ are presented. In particular, for the case of nonsingular leading coefficients of the perturbation matrix polynomial $W(x)$ we present a generalization of the Christoffel formula constructed in terms of the Jordan chains of $W(x)$. For perturbations with a singular leading coefficient several examples by Durán et al are revisited. Finally, we extend these results to the non-Abelian 2D Toda lattice hierarchy.

in International Mathematics Research Notices, May 23, 2016

References in corpus (5)

• Matrix Orthogonal Laurent Polynomials on the Unit Circle and Toda Type Integrable Systems
• Multivariate orthogonal polynomial and integrable systems
• Darboux transformations for multivariate orthogonal polynomials
• On the Christoffel--Darboux formula for generalized matrix orthogonal polynomials of multigraded-Hankel type
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Cited by in corpus (3)

• Multivariate orthogonal polynomial and integrable systems
• Multiple orthogonal polynomials: Pearson equations and Christoffel formulas
• Ladder relations for a class of matrix valued orthogonal polynomials