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Finite‐order meromorphic solutions and the discrete Painlevé equations
Author(s) -
Halburd R. G.,
Korhonen R. J.
Publication year - 2007
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdl012
Subject(s) - meromorphic function , mathematics , order (exchange) , integrable system , riccati equation , mathematical analysis , pure mathematics , differential equation , finance , economics
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.