On the relation between the full Kostant-Toda lattice and multiple orthogonal polynomials

TL;DR

This paper explores the connection between the full Kostant-Toda lattice and multiple orthogonal polynomials, providing explicit solutions and functional representations for the associated integrable systems.

Contribution

It establishes a novel link between the Kostant-Toda lattice and multiple orthogonal polynomials using a Lax pair framework and matrix moments evolution.

Findings

Explicit resolvent function expressions

Representation of the functional vector under certain conditions

Connection between difference operators and integrable systems

Abstract

The correspondence between a high-order non symmetric difference operator with complex coefficients and the evolution of an operator defined by a Lax pair is established. The solution of the discrete dynamical system is studied, giving explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for our integrable systems. The method of investigation is based on the evolutions of the matrical moments.