The multicomponent 2D Toda hierarchy: generalized matrix orthogonal polynomials, multiple orthogonal polynomials and Riemann--Hilbert problems

TL;DR

This paper explores the connection between the multi-component 2D Toda hierarchy and matrix orthogonal polynomials, introducing generalized polynomials, their recursion relations, and links to Riemann--Hilbert problems, advancing integrable systems and orthogonal polynomial theory.

Contribution

It establishes a new relation between multi-component 2D Toda hierarchy, generalized matrix orthogonal polynomials, and Riemann--Hilbert problems, including recursion relations and multiple orthogonal polynomials.

Findings

Derived recursion relations for generalized matrix orthogonal polynomials.

Linked rank one weights to multiple orthogonal polynomials of mixed type.

Connected the hierarchy to Riemann--Hilbert problems for these polynomials.

Abstract

We consider the relation of the multi-component 2D Toda hierarchy with matrix orthogonal and biorthogonal polynomials. The multi-graded Hankel reduction of this hierarchy is considered and the corresponding generalized matrix orthogonal polynomials are studied. In particular for these polynomials we consider the recursion relations, and for rank one weights its relation with multiple orthogonal polynomials of mixed type with a type II normalization and the corresponding link with a Riemann--Hilbert problem.