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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(z+1)+w(z-1)=R(z,w(z)),

where R(z,w(z)) is rational in w(z) 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(z) 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:
39A12Discrete version of topics in analysis
30D35Distribution of values (one complex variable); Nevanlinna theory
34M55Painlevé and other special equations; classification, hierarchies
39A20Generalized difference equations