Finite-order meromorphic solutions and the discrete Painlevé equations.

*(English)*Zbl 1119.39014The authors study the second-order rational difference equation

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

where $R(z,w(z\left)\right)$ 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.

Reviewer: Dingyong Bai (Guangzhou)

##### 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 |