### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 054, 30 pages      arXiv:1511.08098      https://doi.org/10.3842/SIGMA.2016.054
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

### Multidimensional Toda Lattices: Continuous and Discrete Time

Alexander I. Aptekarev a, Maxim Derevyagin b, Hiroshi Miki c and Walter Van Assche d
a) Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, 125047 Moscow, Russia
b) University of Mississippi, Department of Mathematics, Hume Hall 305, P. O. Box 1848, University, MS 38677-1848, USA
c) Doshisha University, Department of Electronics, Faculty of Science and Engineering, Kyotanabe city, Kyoto 610 0394, Japan
d) KU Leuven, Department of Mathematics, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium

Received January 05, 2016, in final form June 01, 2016; Published online June 13, 2016

Abstract
In this paper we present multidimensional analogues of both the continuous- and discrete-time Toda lattices. The integrable systems that we consider here have two or more space coordinates. To construct the systems, we generalize the orthogonal polynomial approach for the continuous and discrete Toda lattices to the case of multiple orthogonal polynomials.

Key words: multiple orthogonal polynomials; orthogonal polynomials; recurrence relations; Toda equation; discrete integrable system; Toda lattice.

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