In this comprehensive survey article the authors firstly describe, by way of examples, some of the remarkable properties of discrete Painlevé equations, such as: the existence of a related linear (iso-monodromy) problem, the existence of Bäcklund transformations and relations to Bäcklund transformations of differential Painlevé equations. Then they present several theorems on the existence of non-trivial meromorphic solutions of certain classes of difference equations: linear difference equations, nonlinear first-order difference equations, the QRT map and some nonlinear higher order difference equations.
An introduction to basic Nevanlinna theory, with some properties of the characteristic function of a meromorphic function and the lemma on the logarithmic derivative are also given. This theory is applied to difference equations admitting finite-order meromorphic solutions. The authors present several strong necessary conditions for an equation to admit a meromorphic solution, by using some different tools: difference analogues of the lemma of the logarithmic derivative, J. Clunie’s lemma [Lond. Math. Soc. 37, 17–27 (1962; Zbl 0104.29504)] and a value distribution result of A. Z. Mokhon’ko and V. D. Mokhon’ko [Sib. Mat. Ab. 15, 1305–1322 (1974; Zbl 0303.30024)]. A new example of an equation of the form , where is rational in with meromorphic coefficients, is studied.
An overview of recent results on meromorphic solutions of linear difference equations, such as: a version of Wiman-Valiron theory for slow growing functions [cf. W. K. Hayman, Can. Math. Bull. 17, 317–358 (1974; Zbl 0314.30021)], order estimates for the growth of finite-order solutions and a theorem concerning minimal solutions of first-order equations, is presented. The authors also present a number of recent results on the value distribution of shifts of finite-order meromorphic functions (difference analogues of the Nevanlinna’s second main theorem, Picard’s theorem, defect relations and theorems concernings meromorphic functions sharing values). They finally describe a -difference analogue of the lemma on the logarithmic derivative, which leads to -difference analogues of some theorems presented in the paper.