Schwarz triangle functions and duality for certain parameters of the generalised Chazy equation

Abstract

Schwarz triangle functions play a fundamental role in the solutions of the generalised Chazy equation. We determine the Schwarz triangle functions that appear in the solutions in the cases where $k=\frac{2}{3}$, $\frac{3}{2}$, $2$ and $3$. Chazy has shown that for the parameters $k=2$ and $3$, the equations can be linearised. Some of the Schwarz triangle functions that show up in the solutions to the generalised Chazy equation with these two parameters also show up in the dual cases where $k=\frac{2}{3}$ and $k=\frac{3}{2}$, suggesting an intriguing connection between the solutions for $k=2$ and $k=3$ with dihedral and tetrahedral symmetry respectively, and the solutions for $k=\frac{2}{3}, \frac{3}{2}$ with $G_2$ symmetry.