An Isomonodromy Interpretation of the Hypergeometric Solution of the Elliptic Painlev\'e Equation (and Generalizations)

TL;DR

This paper develops a framework linking hypergeometric solutions of elliptic Painlevé equations to isomonodromy deformations, introducing new difference equations with biorthogonal function solutions.

Contribution

It constructs a family of difference equations associated with elliptic Painlevé solutions, revealing their isomonodromic deformation structure and biorthogonal function solutions.

Findings

Established a new class of difference equations linked to elliptic Painlevé solutions

Demonstrated monodromy-preserving deformations in these equations

Connected solutions to biorthogonal functions related to elliptic integrals

Abstract

We construct a family of second-order linear difference equations parametrized by the hypergeometric solution of the elliptic Painlev\'e 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.