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
\[ \bar{y}+\underline{y} = R(z,y) \]
with \(R\) being rational in both arguments, \(z\in C\), \(\bar{y}=y(z+1)\), \(\underline{y}=y(z-1)\), \(y=y(z)\). The discrete version of the Painlevé II equation for which
\[ R(z,y)={{(\lambda z+\mu)y+\nu}\over{1-y^2}} \]
with constant parameters \(\lambda,\mu,\nu\) 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.


39A12 Discrete version of topics in analysis
39A20 Multiplicative and other generalized difference equations
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
Full Text: DOI


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