Discrete integrable systems generated by Hermite-Padé approximants

Abstract

We consider Hermite-Padé 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-Padé approximants exists. In addition, we give a few algorithms to find solutions of the discrete system.

20 pages

References in corpus (1)

• Computing recurrence coefficients of multiple orthogonal polynomials

Cited by in corpus (3)

• Non-commutative Hermite--Padé approximation and integrability
• Non-autonomous multidimensional Toda system and multiple interpolation problem
• Christoffel Transform and Multiple Orthogonal Polynomials