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.
- Publication:
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arXiv e-prints
- Pub Date:
- September 2014
- DOI:
- 10.48550/arXiv.1409.4053
- arXiv:
- arXiv:1409.4053
- Bibcode:
- 2014arXiv1409.4053A
- Keywords:
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- Mathematics - Classical Analysis and ODEs;
- 42C05;
- 37K10
- E-Print:
- 20 pages