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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:

39A12 Discrete version of topics in analysis
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
39A20 Multiplicative and other generalized difference equations
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