On the tau function of the hypergeometric equation

TL;DR

This paper defines the tau-function for a rank-two hypergeometric system with three singularities, expressing it explicitly via Barnes G-functions and linking it to Painlevé VI solutions.

Contribution

It introduces a new formulation of the tau-function as a generating function for monodromy symplectomorphism, explicitly relating it to monodromy data and Barnes G-functions.

Findings

Explicit tau-function in terms of Barnes G-function

Connection to Painlevé VI asymptotics

Determination of tau-function dependence on monodromy data

Abstract

The monodromy map for a rank-two system of differential equations with three Fuchsian singularities is classically solved by the Kummer formul\ae\ for Gauss' hypergeometric functions. We define the tau-function of such a system as the generating function of the extended monodromy symplectomorphism, using an idea recently developed. This formulation allows us to determine the dependence of the tau-function on the monodromy data. Using the explicit solution of the monodromy problem, the tau-function is then explicitly written in terms of Barnes $G$-function. In particular, if the Fuchsian singularities are placed to $0$, $1$ and $\infty$, this gives the structure constants of the asymptotical formula of Iorgov-Gamayun-Lisovyy for solutions of Painlev\'e VI equation.