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On the extension of the Painlevé property to difference equations. (English) Zbl 0956.39003

The authors find some conditions under which a difference equation would be considered to be of Painlevé type. They are led to the following conclusions.

1) Arbitrary periodic functions play an analogous role in the solutions of difference equations to that played by constants in the solutions to ordinary differential equations and arbitrary functions on characteristic functions on characteristic manifolds in solutions to partial differential equations.

2) The solutions to many difference equations can be extended to the complex plane and are meromorphic.

3) The integrability of a large class of difference equations is associated with the asymptotic structure of its solutions at infinity.

4) Nevanlinna theory provides many tools that can be used to analyse the integrability of some equations.

5) Expansions of solutions can be constructed using perturbative techniques which are difference versions of those used by Painlevé for differential equations.


MSC:
39A10Additive difference equations
34M55Painlevé and other special equations; classification, hierarchies
39A12Discrete version of topics in analysis
30D35Distribution of values (one complex variable); Nevanlinna theory