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arXiv · 1409.4053

Discrete integrable systems generated by Hermite-Padé approximants

Alexander I. Aptekarev·Maxim Derevyagin·Walter Van Assche
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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.

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Alexander I. Aptekarev, Maxim Derevyagin, Walter Van Assche. 2015-09-29. Discrete integrable systems generated by Hermite-Padé approximants. https://doi.org/10.1088/0951-7715%2F29%2F5%2F1487

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