Abstract
Let w(z) be an admissible finite-order meromorphic solution of the second-order 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. 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 suggests that the existence of finite-order meromorphic solutions is a good detector of integrable difference equations.
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