Singularity confinement for a class of $m$-th order difference equations of combinatorics

Abstract

In a recent publication, it was shown that a large class of integrals over the unitary group U(n) satisfy difference equations over $n$, involving a finite number of steps; special cases are generating functions appearing in questions of longest increasing subsequences in random permutations and words. The main result of the paper states that these difference equations have the \emph{discrete Painlev\'e property}; roughly speaking, this means that, after a finite number of steps, the solution to these difference equations may develop a pole (Laurent solution), depending on the maximal number of free parameters, and immediately after be finite again (``\emph{singularity confinement}''). The technique used in the proof is based on an intimate relationship between the difference equations (discrete time) and the Toeplitz lattice (continuous time differential equations); the point is that the ``Painlev\'e property'' for the discrete relations is inherited from the ``Painlev\'e property'' of the (continuous) Toeplitz lattice.