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SIGMA, 2012, Volume 8, 097, 27 pages (Mi sigma774)  

This article is cited in 7 scientific papers (total in 7 papers)

Construction of a Lax Pair for the $E_6^{(1)}$ $q$-Painlevé System

Nicholas S. Wittea, Christopher M. Ormerodb

a Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia
b Department of Mathematics and Statistics, La Trobe University, Bundoora VIC 3086, Australia

Abstract: We construct a Lax pair for the $ E^{(1)}_6 $ $q$-Painlevé system from first principles by employing the general theory of semi-classical orthogonal polynomial systems characterised by divided-difference operators on discrete, quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such lattices – the $q$-linear lattice – through a natural generalisation of the big $q$-Jacobi weight. As a by-product of our construction we derive the coupled first-order $q$-difference equations for the $ E^{(1)}_6 $ $q$-Painlevé system, thus verifying our identification. Finally we establish the correspondences of our result with the Lax pairs given earlier and separately by Sakai and Yamada, through explicit transformations.

Keywords: non-uniform lattices; divided-difference operators; orthogonal polynomials; semi-classical weights; isomonodromic deformations; Askey table

DOI: https://doi.org/10.3842/SIGMA.2012.097

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Full text: http://emis.mi.ras.ru/.../097
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Bibliographic databases:

ArXiv: 1207.0041
MSC: 39A05; 42C05; 34M55; 34M56; 33C45; 37K35
Received: September 5, 2012; in final form November 29, 2012; Published online December 11, 2012
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Citation: Nicholas S. Witte, Christopher M. Ormerod, “Construction of a Lax Pair for the $E_6^{(1)}$ $q$-Painlevé System”, SIGMA, 8 (2012), 097, 27 pp.

Citation in format AMSBIB
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\by Nicholas~S.~Witte, Christopher~M.~Ormerod
\paper Construction of a Lax Pair for the $E_6^{(1)}$ $q$-Painlev\'e System
\jour SIGMA
\yr 2012
\vol 8
\papernumber 097
\totalpages 27
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. C. M. Ormerod, P. H. van der Kamp, G. R. W. Quispel, “Discrete Painlevé equations and their Lax pairs as reductions of integrable lattice equations”, J. Phys. A-Math. Theor., 46:9 (2013), 095204  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Christopher M. Ormerod, “Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation”, SIGMA, 10 (2014), 002, 19 pp.  mathnet  crossref  mathscinet
    3. Nagao H., “the Pad, Interpolation Method Applied To Q-Painlev, Equations”, Lett. Math. Phys., 105:4 (2015), 503–521  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. Christopher M. Ormerod, Eric M. Rains, “Commutation Relations and Discrete Garnier Systems”, SIGMA, 12 (2016), 110, 50 pp.  mathnet  crossref
    5. Joshi N. Nakazono N. Shi Ya., “Lattice equations arising from discrete Painlevé systems: II. ${A}_{4}^{(1)}$ case”, J. Phys. A-Math. Theor., 49:49 (2016), 495201  crossref  mathscinet  zmath  isi  scopus
    6. Joshi N., Nakazono N., “Lax pairs of discrete Painlevé equations: $(A_2+A_1)^{(1)}$ case”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 472:2196 (2016), 20160696  crossref  mathscinet  zmath  isi  scopus
    7. K. Kajiwara, M. Noumi, Ya. Yamada, “Geometric aspects of Painlevé equations”, J. Phys. A-Math. Theor., 50:7 (2017), 073001  crossref  mathscinet  zmath  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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