A Tau function for $q$-Painlev\'e VI as a Fredholm determinant

Abstract

We give an analytic construction of a tau function for the $q$-difference sixth Painlev\'e equation ($q$PVI) as a Fredholm determinant through the general Riemann-Hilbert problem associated with it. We show that the tau function is an analytic function on its domain of definition that vanishes at a particular point if and only if the corresponding Riemann-Hilbert problem is point-wise not solvable there. We express the corresponding $q$PVI transcendents in terms of the tau function as well as three copies of it with some of the parameters shifted. Then the vanishing of any of these four tau functions corresponds to the transcendents taking value in a specific corresponding exceptional line on the initial value space of $q$PVI. Finally, we derive an asymptotic expansion of the tau function for small times $t$.