The theme running throughout this thesis is the Painlevé equations, in their differential,
discrete and ultra-discrete versions. The differential Painlevé equations have
the Painlevé property. If all solutions of a differential equation are meromorphic
functions then it necessarily has the Painlevé property. Any ODE with the Painlevé
property is necessarily a reduction of an integrable PDE.
Nevanlinna theory studies the value distribution and characterizes the growth
of meromorphic functions, by using certain averaged properties on a disc of variable
radius. We shall be interested in its well-known use as a tool for detecting
integrability in difference equations—a difference equation may be integrable if it
has sufficiently many finite-order solutions in the sense of Nevanlinna theory. This
does not provide a sufficient test for integrability; additionally it must satisfy the
well-known singularity confinement test. [Continues.
