Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations

The Painlevé property is closely connected to differential equations that are integrable via related iso-monodromy problems. Many apparently integrable discrete analogues of the Painlevé equations have appeared in the literature. The existence of sufficiently many finite-order meromorphic solutions appears to be a good analogue of the Painlevé property for discrete equations, in which the independent variable is taken to be complex. A general introduction to Nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q-difference operators. New results are presented concerning equations of the form w(z + 1)w(z− 1) = R(z,w), where R is rational in w with meromorphic coefficients. PACS numbers: 02.30.Ik, 02.30.Ks, 05.45.−a