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Existence of finite-order meromorphic solutions as a detector of integrability in difference equations. (English) Zbl 1105.39019

The authors consider an analogue of the Painlevé property for discrete equations. They start from a difference equations of the type

y ¯+y ̲=R(z,y)

with R being rational in both arguments, zC, y ¯=y(z+1), y ̲=y(z-1), y=y(z). The discrete version of the Painlevé II equation for which

R(z,y)=(λz+μ)y+ν 1-y 2

with constant parameters λ,μ,ν belongs to this class of equations. For this equation the “integrability” test based on singularity confinement for meromorphic solutions is applied; this confinement is discussed using the Nevanlinna theory.

MSC:
39A12Discrete version of topics in analysis
39A20Generalized difference equations
34M55Painlevé and other special equations; classification, hierarchies
37F10Polynomials; rational maps; entire and meromorphic functions