Hypergeometric solution of a certain polynomial Hamiltonian system of isomonodromy type

TL;DR

This paper presents particular solutions for a class of polynomial Hamiltonian systems related to isomonodromy problems, using generalized hypergeometric functions, Pfaffian systems, and Lax formalism.

Contribution

It introduces explicit solutions of polynomial Hamiltonian systems via a generalized hypergeometric function framework, connecting integral representations and Lax formalism.

Findings

Solutions expressed in terms of generalized hypergeometric functions

Connection between Pfaffian systems and Hamiltonian solutions

Extension of classical hypergeometric functions to isomonodromy systems

Abstract

In our previous work, a unified description as polynomial Hamiltonian systems was established for a broad class of the Schlesinger systems including the sixth Painleve equation and Garnier systems. The main purpose of this paper is to present particular solutions of this Hamiltonian system in terms of a certain generalization of Gauss' hypergeometric function. Key ingredients of the argument are the linear Pfaffian system derived from an integral representation of the hypergeometric function (with the aid of twisted de Rham theory) and Lax formalism of the Hamiltonian system.