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Proceedings of the London Mathematical Society Advance Access published online on December 6, 2006
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdl012
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© 2006 London Mathematical Society

Finite-order meromorphic solutions and the discrete Painlevé equations

R. G. Halburd and R. J. Korhonen

Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom R.G.Halburd{at}lboro.ac.uk
Department of Mathematics, University of Joensuu, P.O. Box 111, FI-80101 Joensuu, Finland risto.korhonen{at}joensuu.fi

Received 24 February 2005. Revision received 24 April 2006.

Let w(z) be an admissible finite-order meromorphic solution of the second-order difference equation

Formula 012UM1
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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