Non-commutative Hermite--Pad\'{e} approximation and integrability

TL;DR

This paper develops a non-commutative Hermite-Padé approximation framework, linking it to integrable systems like the non-commutative discrete Toda lattice, and demonstrates the integrability of these systems using quasideterminants.

Contribution

It introduces a non-commutative Hermite-Padé approximation method and connects it to integrable non-commutative systems, providing new solutions and proving their integrability.

Findings

Solution expressed by quasideterminants.

Connection to non-commutative Hirota system.

Proof of integrability of the non-commutative Toda lattice.

Abstract

We introduce and solve the non-commutative version of the Hermite-Pad\'{e} type I approximation problem. Its solution, expressed by quasideterminants, leads in a natural way to a subclass of solutions of the non-commutative Hirota (discrete Kadomtsev--Petviashvili) system and of its linear problem. We also prove integrability of the constrained system, which in the simplest case is the non-commutative discrete-time Toda lattice equation known from the theory of non-commutative Pad\'{e} approximants and matrix orthogonal polynomials.