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Zbl 1119.39014
Halburd, R.G.; Korhonen, R.J.
Finite-order meromorphic solutions and the discrete Painlevé equations. (English)
[J] Proc. Lond. Math. Soc. (3) 94, No. 2, 443-474 (2007). ISSN 0024-6115; ISSN 1460-244X/e

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.
[Dingyong Bai (Guangzhou)]

MSC 2000:: *39A12 Discrete version of topics in analysis
30D35 Distribution of values (one complex variable)
34M55 Painlevé and other special equations
39A20 Generalized difference equations

Keywords: second-order rational difference equation; meromorphic solution; difference Painlevé equations; integrability

Cited in: Zbl 1219.39001 Zbl 1180.30039

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