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

SIGMA 8 (2012), 008, 14 pages      arXiv:1111.7262      https://doi.org/10.3842/SIGMA.2012.008

### Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems

Hiroshi Miki, Hiroaki Goda and Satoshi Tsujimoto
Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Sakyo-Ku, Kyoto 606 8501, Japan

Received December 01, 2011, in final form February 20, 2012; Published online February 29, 2012

Abstract
Discrete spectral transformations of skew orthogonal polynomials are presented. From these spectral transformations, it is shown that the corresponding discrete integrable systems are derived both in 1+1 dimension and in 2+1 dimension. Especially in the (2+1)-dimensional case, the corresponding system can be extended to 2×2 matrix form. The factorization theorem of the Christoffel kernel for skew orthogonal polynomials in random matrix theory is presented as a by-product of these transformations.

Key words: skew orthogonal polynomials; discrete integrable systems; discrete coupled KP equation; Pfaff lattice; Christoffel-Darboux kernel.

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