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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 208701, 9 pages
http://dx.doi.org/10.1155/2014/208701
Research Article

On Properties of Meromorphic Solutions of Certain Difference Painlevé III Equations

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received 5 November 2013; Accepted 13 January 2014; Published 27 February 2014

Academic Editor: Kwang Ho Shon

Copyright © 2014 Shuang-Ting Lan and Zong-Xuan Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, UK, 1964. View at MathSciNet
  2. L. Yang, Value Distribution Theory and Its New Research, Science Press, Beijing, China, 1982, (Chinese).
  3. L. Fuchs, “Sur quelques équations différentielles linéares du second ordre,” Comptes Rendus de l'Académie des Sciences, vol. 141, pp. 555–558, 1905.
  4. B. Gambier, “Sur les équations différentielles du second ordre et du premier degré dont l'intégrale générale est a points critiques fixes,” Acta Mathematica, vol. 33, no. 1, pp. 1–55, 1910. View at Publisher · View at Google Scholar · View at MathSciNet
  5. P. Painlevé, “Mémoire sur les équations différentielles dont l'intégrale générale est uniforme,” Bulletin de la Société Mathématique de France, vol. 28, pp. 201–261, 1900. View at MathSciNet
  6. P. Painlevé, “Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme,” Acta Mathematica, vol. 25, no. 1, pp. 1–85, 1902. View at Publisher · View at Google Scholar · View at MathSciNet
  7. M. J. Ablowitz and H. Segur, “Exact linearization of a Painlevé transcendent,” Physical Review Letters, vol. 38, no. 20, pp. 1103–1106, 1977. View at Publisher · View at Google Scholar · View at MathSciNet
  8. B. Q. Chen, Z. X. Chen, and S. Li, “Uniqueness theorems on entire functions and their difference operators or shifts,” Abstract and Applied Analysis, vol. 2012, Article ID 906893, 8 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Y.-M. Chiang and S.-J. Feng, “On the Nevanlinna characteristic of f(z+η) and difference equations in the complex plane,” Ramanujan Journal, vol. 16, no. 1, pp. 105–129, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. R. G. Halburd and R. J. Korhonen, “Difference analogue of the lemma on the logarithmic derivative with applications to difference equations,” Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 477–487, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. I. Laine and C.-C. Yang, “Clunie theorems for difference and q-difference polynomials,” Journal of the London Mathematical Society, vol. 76, no. 3, pp. 556–566, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  12. R. R. Zhang and Z. B. Huang, “Results on difference analogues of Valiron-Mohon'ko theorem,” Abstract and Applied Analysis, vol. 2013, Article ID 273040, 6 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. J. Ablowitz, R. Halburd, and B. Herbst, “On the extension of the Painlevé property to difference equations,” Nonlinearity, vol. 13, no. 3, pp. 889–905, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. R. G. Halburd and R. J. Korhonen, “Finite-order meromorphic solutions and the discrete Painlevé equations,” Proceedings of the London Mathematical Society, vol. 94, no. 2, pp. 443–474, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Z.-X. Chen and K. H. Shon, “Value distribution of meromorphic solutions of certain difference Painlevé equations,” Journal of Mathematical Analysis and Applications, vol. 364, no. 2, pp. 556–566, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. O. Ronkainen, “Meromorphic solutions of difference Painlevé equations,” Academiæ Scientiarum Fennicæ. Annales. Mathematica. Dissertationes, no. 155, p. 59, 2010. View at Zentralblatt MATH · View at MathSciNet
  17. J. L. Zhang and L. Z. Yang, “Meromorphic solutions of Painlevé III difference equations,” Acta Mathematica Sinica, vol. 57, no. 1, pp. 181–188, 2014.
  18. A. Z. Mokhon’ko, “On the Nevanlinna characteristics of some meromorphic functions,” in Theory of Functions, Functional Analysis and Their Applications, vol. 14, pp. 83–87, I zd-vo Khar’kovsk. Un-ta, 1971.