Discrete integrable systems generated by Hermite-Pad\'e approximants

TL;DR

This paper explores the connection between Hermite-Padé approximants and discrete integrable systems, revealing a bidirectional relationship and providing algorithms for solutions, advancing understanding in approximation theory and integrable systems.

Contribution

It establishes a deep link between multiple orthogonality and discrete integrable systems, and demonstrates the existence of perfect systems for given integrable systems.

Findings

Hermite-Padé approximants relate to Lax representations

Discrete integrable systems correspond to perfect systems

Algorithms for solving the systems are provided

Abstract

We consider Hermite-Pad\'e approximants in the framework of discrete integrable systems defined on the lattice $\mathbb{Z}^2$. We show that the concept of multiple orthogonality is intimately related to the Lax representations for the entries of the nearest neighbor recurrence relations and it thus gives rise to a discrete integrable system. We show that the converse statement is also true. More precisely, given the discrete integrable system in question there exists a perfect system of two functions, i.e., a system for which the entire table of Hermite-Pad\'e approximants exists. In addition, we give a few algorithms to find solutions of the discrete system.