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

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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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
\Bibitem{Rai11} \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 \mathnet{http://mi.mathnet.ru/sigma646} \crossref{https://doi.org/10.3842/SIGMA.2011.088} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2861188} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000294717500004} 

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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.
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
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
4. Christopher M. Ormerod, Eric M. Rains, “Commutation Relations and Discrete Garnier Systems”, SIGMA, 12 (2016), 110, 50 pp.
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
6. Kajiwara K. Noumi M. Yamada Ya., “Geometric aspects of Painlevé equations”, J. Phys. A-Math. Theor., 50:7 (2017), 073001
7. Yasuhiko Yamada, “An Elliptic Garnier System from Interpolation”, SIGMA, 13 (2017), 069, 8 pp.
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.
9. Ormerod Ch.M. Rains E.M., “An Elliptic Garnier System”, Commun. Math. Phys., 355:2 (2017), 741–766
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
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
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