A q-discrete Analogue of the Third Painlevé Equation and its Linear Problem

Abstract

In this thesis we investigate the rational and Riccati type special solutions for particular parameter values of a q-discrete analogue of the third Painlevé equation, with rational surface A(1) 5 and affine Weyl group (A2 + A1)(1). The general solutions of this equation are highly transcendental in nature. We work closely with an associated system of discrete linear equations, which we refer to as ‘the linear problem.’We demonstrate that the linear problem can be solved both in terms of q-Gamma functions and series expansions for different parameter values of our discrete Painlevé equation. By developing a Schlesinger transformation, which transforms a series expansion of the linear problem for one parameter value to another series expansion of the linear problem for another parameter value, we are able to develop determinantal representations of the rational and Riccati type special solutions. These determinantal forms appear different to those discovered previously by Kajiwara. This technique has only been used to develop the determinantal forms of two other continuous and discrete Painlevé equations and hence the results presented here further indicate its potential.