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

SIGMA 7 (2011), 088, 24 pages      arXiv:0807.0258      https://doi.org/10.3842/SIGMA.2011.088
Contribution to the Special Issue “Relationship of Orthogonal Polynomials and Special Functions with Quantum Groups and Integrable Systems”

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

Received April 25, 2011, in final form September 06, 2011; Published online September 09, 2011

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.

Key words: isomonodromy; hypergeometric; Painlevé; biorthogonal functions.

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