*(English)*Zbl 1115.39024

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 $w(z+1)w(z-1)=R(z,w(z\left)\right)$, where $R$ is rational in $w$ 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 $q$-difference analogue of the lemma on the logarithmic derivative, which leads to $q$-difference analogues of some theorems presented in the paper.

##### MSC:

39A13 | Difference equations, scaling ($q$-differences) |

34M55 | Painlevé and other special equations; classification, hierarchies |

30D35 | Distribution of values (one complex variable); Nevanlinna theory |