Discrete integrable systems generated by HermitePadé approximants
Abstract
We consider HermitePadé 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 HermitePadé approximants exists. In addition, we give a few algorithms to find solutions of the discrete system.
 Publication:

arXiv eprints
 Pub Date:
 September 2014
 arXiv:
 arXiv:1409.4053
 Bibcode:
 2014arXiv1409.4053A
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 42C05;
 37K10
 EPrint:
 20 pages