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Meromorphic solutions of difference equations, integrability and the discrete Painlevé equations. (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, {\it J. Clunie}’s lemma [Lond. Math. Soc. 37, 17--27 (1962; Zbl 0104.29504)] and a value distribution result of {\it A. Z. Mokhon’ko} and {\it V. D. Mokhon’ko} [Sib. Mat. Ab. 15, 1305--1322 (1974; Zbl 0303.30024)]. A new example of an equation of the form $w(z+1)w(z-1)=R(z,w(z))$, 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. {\it 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
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