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 SIGMA, 2012, Volume 8, 008, 14 pages (Mi sigma685)

Discrete spectral transformations of skew orthogonal polynomials and associated discrete integrable systems

Hiroshi Miki, Hiroaki Goda, Satoshi Tsujimoto

Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Sakyo-Ku, Kyoto 606 8501, Japan

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 $2\times 2$ 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.

Keywords: skew orthogonal polynomials, discrete integrable systems, discrete coupled KP equation, Pfaff lattice, Christoffel–Darboux kernel.

DOI: https://doi.org/10.3842/SIGMA.2012.008

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Bibliographic databases:

ArXiv: 1111.7262
MSC: 42C05; 35C05; 37K60; 15B52
Received: December 1, 2011; in final form February 20, 2012; Published online February 29, 2012
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Citation: Hiroshi Miki, Hiroaki Goda, Satoshi Tsujimoto, “Discrete spectral transformations of skew orthogonal polynomials and associated discrete integrable systems”, SIGMA, 8 (2012), 008, 14 pp.

Citation in format AMSBIB
\Bibitem{MikGodTsu12}
\by Hiroshi Miki, Hiroaki Goda, Satoshi Tsujimoto
\paper Discrete spectral transformations of skew orthogonal polynomials and associated discrete integrable systems
\jour SIGMA
\yr 2012
\vol 8
\papernumber 008
\totalpages 14
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\crossref{https://doi.org/10.3842/SIGMA.2012.008}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2900507}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84858984612}

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This publication is cited in the following articles:
1. Kazuki Maeda, Satoshi Tsujimoto, “Direct Connection between the $R_{II}$ Chain and the Nonautonomous Discrete Modified KdV Lattice”, SIGMA, 9 (2013), 073, 12 pp.
2. X.-B. Hu, Sh.-H. Li, “The partition function of the bures ensemble as the tau-function of BKP and DKP hierarchies: continuous and discrete”, J. Phys. A-Math. Theor., 50:28 (2017), 285201
3. K. Maeda, “Nonautonomous ultradiscrete hungry Toda lattice and a generalized box-ball system”, J. Phys. A-Math. Theor., 50:36 (2017), 365204
4. X.-K. Chang, Y. He, X.-B. Hu, Sh.-H. Li, “A new integrable convergence acceleration algorithm for computing Brezinski–durbin–redivo–zaglia's sequence transformation via Pfaffians”, Numer. Algorithms, 78:1 (2018), 87–106
5. X. Chang, Y. He, X. Hu, Sh. Li, H.-W. Tam, Y. Zhang, “Coupled modified KdV equations, skew orthogonal polynomials, convergence acceleration algorithms and Laurent property”, Sci. China-Math., 61:6 (2018), 1063–1078
6. B. Wang, X.-K. Chang, X.-B. Hu, Sh.-H. Li, “On moving frames and Toda lattices of BKP and CKP types”, J. Phys. A-Math. Theor., 51:32 (2018), 324002
7. Chang X.-K., He Y., Hu X.-B., Li Sh.-H., “Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices With Pfaffian Tau-Functions”, Commun. Math. Phys., 364:3 (2018), 1069–1119
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