Uniqueness of Meromorphic Functions Concerning Their Differences and Solutions of Difference PainlevÉ Equations

  • Xiaoguang Qi
  • Nan Li
  • Lianzhong Yang


This paper is devoted to studying some shared value properties for finite-order meromorphic solutions of the difference Painlevé IV equation. We also consider sharing value problems for the derivative of a meromorphic function f(z) with its shift \(f(z+c)\) and difference \(\Delta f\).


Meromorphic functions Difference Painlevé equation Value sharing 

Mathematics Subject Classification

Primary 39A05 Secondary 30D35 



The authors would like to thank the referee for his/her helpful suggestions and comments.


  1. 1.
    Charak, K.S., Korhonen, R.J., Kumar, G.: A note on partial sharing of values of meromorphic functions with their shifts. J. Math. Anal. Appl. 435, 1241–1248 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, Z.X., Yi, H.X.: On sharing values of meromorphic functions and their differences. Results Math. 63, 557–565 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chiang, Y.M., Feng, S.J.: On the Nevanlinna characteristic of \(f(z+\eta )\) and difference equations in the complex plane. Ramanujan J. 16, 105–129 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bergweiler, W., Langley, J.K.: Zeros of differences of meromorphic functions. Math. Proc. Camb. Philos. Soc. 142, 133–147 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gundersen, G.G.: Meromorphic functions that share three values IM and a fourth value CM. Complex Var. Elliptic Equ. 20, 99–106 (1992)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Halburd, R.G., Korhonen, R.J.: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. 31, 463–478 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Halburd, R.G., Korhonen, R.J.: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 314, 477–487 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Halburd, R.G., Korhonen, R.J.: Finite order solutions and the discrete Painlevé equations. Proc. Lond. Math. Soc. 94, 443–474 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Heittokangas, J., Korhonen, R., Laine, I., Rieppo, J., Zhang, J.L.: Value sharing results for shifts of meromorphic function, and sufficient conditions for periodicity. J. Math. Anal. Appl. 355, 352–363 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Heittokangas, J., Korhonen, R., Laine, I., Rieppo, J.: Uniqueness of meromorphic functions sharing values with their shifts. Complex Var. Elliptic Equ. 56, 81–92 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lin, W.C., Tohge, K.: On shared-value properties of Painlevé transcendents. Comput. Methods Funct. Theory 7, 477–499 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lü, F., Han, Q., Lü, W.R.: On unicity of meromorphic solutions to difference equations of Malmquist type. Bull. Aust. Math. Soc. 93, 92–98 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lü, F., Lü, W.R.: Meromorphic functions sharing three values with their difference operators. Comput. Methods Funct. Theory 17, 395–403 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Qi, X.G.: Value distribution and uniqueness of difference polynomials and entire solutions of difference equations. Ann. Polon. Math. 102, 129–142 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Qi, X.G., Dou, J.: On shared value properties of difference Painlevé equations, submittedGoogle Scholar
  16. 16.
    Ronkainen, O.: Meromorphic solutions of difference Painlevé equations. Doctoral Dissertation, Helsinki (2010)Google Scholar
  17. 17.
    Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Kluwer Academic Publishers, Dordrecht (2003)CrossRefzbMATHGoogle Scholar
  18. 18.
    Yang, L.Z., Zhang, J.L.: Non-existence of meromorphic solution of a Fermat type functional equation. Aequ. Math. 76, 140–150 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Zhang, J.L.: Meromorphic solutions of Painlevé IV difference equations. Adv. Differ. Equ. 260, (2014)Google Scholar
  20. 20.
    Zhang, J., Liao, L.W.: Entire functions sharing some values with their difference operators. Sci China Math. 57, 2143–2152 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsUniversity of JinanJinanPeople’s Republic of China
  2. 2.School of MathematicsShandong UniversityJinanPeople’s Republic of China

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