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Finite-order meromorphic solutions and the discrete Painlevé equations. (English) Zbl 1119.39014

The authors study the second-order rational difference equation

$w\left(z+1\right)+w\left(z-1\right)=R\left(z,w\left(z\right)\right),$

where $R\left(z,w\left(z\right)\right)$ is rational in $w\left(z\right)$ with coefficients that are meromorphic in $z$. They show that if the equation has at least one admissible meromorphic solution of finite order, then either $w\left(z\right)$ satisfies a difference linear or Riccati equation or else the above equation can be transformed to one of a list of canonical difference equations. This list consists of all known difference Painlevé equations of the above form, together with their autonomous versions. This indicates that the existence of a finite-order meromorphic solution of a difference equation is a strong indicator of integrability of the equation.

##### MSC:
 39A12 Discrete version of topics in analysis 30D35 Distribution of values (one complex variable); Nevanlinna theory 34M55 Painlevé and other special equations; classification, hierarchies 39A20 Generalized difference equations