Discrete integrable systems generated by Hermite-Pade approximants

We consider Hermite-Pade approximants in the framework of discrete integrable systems defined on the lattice 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-Pade approximants exists. In addition, we give a few algorithms to find solutions of the discrete system.