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An Isaac Newton Institute Workshop

Painleve Equations and Monodromy Problems: Recent Developments

Finite-order meromorphic solutions and the discrete Painleve equations

Authors: R. G. Halburd (Loughborough University), R. J. Korhonen (University of Joensuu)

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 form w(z+1)+w(z-1)=R(z,w(z)), together with their autonomous versions. This suggests that the existence of finite-order meromorphic solutions is a good detector of integrable difference equations.