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SIGMA, 2011, Volume 7, 088, 24 pages (Mi sigma646)  

This article is cited in 11 scientific papers (total in 11 papers)

An Isomonodromy Interpretation of the Hypergeometric Solution of the Elliptic Painlevé Equation (and Generalizations)

Eric M. Rains

Department of Mathematics, California Institute of Technology, 1200 E. California Boulevard, Pasadena, CA 91125, USA

Abstract: We construct a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painlevé equation (or higher-order analogues), and admitting a large family of monodromy-preserving deformations. The solutions are certain semiclassical biorthogonal functions (and their Cauchy transforms), biorthogonal with respect to higher-order analogues of Spiridonov's elliptic beta integral.

Keywords: isomonodromy; hypergeometric; Painlevé; biorthogonal functions

DOI: https://doi.org/10.3842/SIGMA.2011.088

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Full text: http://emis.mi.ras.ru/.../088
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Bibliographic databases:

ArXiv: 0807.0258
MSC: 33E17; 34M55; 39A13
Received: April 25, 2011; in final form September 6, 2011; Published online September 9, 2011
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Citation: Eric M. Rains, “An Isomonodromy Interpretation of the Hypergeometric Solution of the Elliptic Painlevé Equation (and Generalizations)”, SIGMA, 7 (2011), 088, 24 pp.

Citation in format AMSBIB
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\by Eric M. Rains
\paper An Isomonodromy Interpretation of the Hypergeometric Solution of the Elliptic Painlev\'e Equation (and Generalizations)
\jour SIGMA
\yr 2011
\vol 7
\papernumber 088
\totalpages 24
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\crossref{https://doi.org/10.3842/SIGMA.2011.088}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2861188}
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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. Christopher M. Ormerod, “Symmetries and Special Solutions of Reductions of the Lattice Potential KdV Equation”, SIGMA, 10 (2014), 002, 19 pp.  mathnet  crossref  mathscinet
    2. Witte N.S., “Semiclassical Orthogonal Polynomial Systems on Nonuniform Lattices, Deformations of the Askey Table, and Analogues of Isomonodromy”, Nagoya Math. J., 219 (2015), 127–234  crossref  mathscinet  zmath  isi  elib  scopus
    3. Ramani A. Grammaticos B., “Discrete Painlevé Equations Associated With the Affine Weyl Group E-8”, J. Phys. A-Math. Theor., 48:35 (2015), 355204  crossref  mathscinet  zmath  isi  scopus
    4. Christopher M. Ormerod, Eric M. Rains, “Commutation Relations and Discrete Garnier Systems”, SIGMA, 12 (2016), 110, 50 pp.  mathnet  crossref
    5. Hietarinta J., Joshi N., Nijhoff F., “Discrete Systems and Integrability”, Discrete Systems and Integrability, Cambridge Texts in Applied Mathematics, Cambridge Univ Press, 2016, 1–445  mathscinet  zmath  isi
    6. Kajiwara K. Noumi M. Yamada Ya., “Geometric aspects of Painlevé equations”, J. Phys. A-Math. Theor., 50:7 (2017), 073001  crossref  mathscinet  zmath  isi  scopus
    7. Yasuhiko Yamada, “An Elliptic Garnier System from Interpolation”, SIGMA, 13 (2017), 069, 8 pp.  mathnet  crossref
    8. Hidehito Nagao, “A Variation of the $q$-Painlevé System with Affine Weyl Group Symmetry of Type $E_7^{(1)}$”, SIGMA, 13 (2017), 092, 18 pp.  mathnet  crossref
    9. Ormerod Ch.M. Rains E.M., “An Elliptic Garnier System”, Commun. Math. Phys., 355:2 (2017), 741–766  crossref  mathscinet  zmath  isi  scopus
    10. Noumi M., “Remarks on Tau-Functions For the Difference Painleve Equations of Type E-8”, Representation Theory, Special Functions and Painleve Equations - Rims 2015, Advanced Studies in Pure Mathematics, 76, eds. Konno H., Sakai H., Shiraishi J., Suzuki T., Yamada Y., Math Soc Japan, 2018, 1–65  mathscinet  isi
    11. Nijhoff F., Delice N., “On Elliptic Lax Pairs and Isomonodromic Deformation Systems For Elliptic Lattice Equations in Honour of Professor Noumi For the Occasion of His 60Th Birthday”, Representation Theory, Special Functions and Painleve Equations - Rims 2015, Advanced Studies in Pure Mathematics, 76, eds. Konno H., Sakai H., Shiraishi J., Suzuki T., Yamada Y., Math Soc Japan, 2018, 487–525  mathscinet  isi
  • Symmetry, Integrability and Geometry: Methods and Applications
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