Skip to main content
Log in

Meromorphic Solutions of Complex Differential–Difference Equations

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

This paper is devoted to exploring the properties of meromorphic solutions on complex differential–difference equations using Nevanlinna theory. We state some relationships between the exponent of convergence of zeros with the order of meromorphic solutions on linear or non-linear differential–difference equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Chen, Z.X.: Zeros of entire solutions to complex linear difference equations. Acta Math. Sci. Ser. B. Engl. Ed. 32(2), 1141–1148 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Chen, Z.X.: Growth and zeros of meromorphic solution of some linear difference equations. J. Math. Anal. Appl. 373, 235–241 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chen, Z.X.: Complex Differences and Difference Equations, Mathematics Monograph Series 29. Science Press, Beijing (2014)

  4. Chen, Z.X., Yang, C.C.: On entire solutions of certain type of differential–difference equations. Taiwan. J. Math. 18(3), 677–685 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Halburd, R.G., Korhonen, R.J.: Value distribution and linear operators. Proc. Edinb. Math. Soc. 57(2), 493–504 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Halburd, R.G., Korhonen, R.J.: Growth of meromorphic solutions of delay differential equation. Proc. Am. Math. Soc. 145, 2513–2526 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hayman, W.K.: Meromorphic Functions. Clarendon Press, Oxford (1964)

    MATH  Google Scholar 

  10. Hurwitz, A.: Mathematische Werke, vol. II. Birkhäuser, Basel (1933)

    MATH  Google Scholar 

  11. Laine, I.: Nevanlinna Theory and Complex Differential Equations. Walter de Gruyter, Berlin, New York (1993)

    Book  MATH  Google Scholar 

  12. Liu, K., Dong, X.J.: Some results related to complex differential–difference equations of certain types. Bull. Korean Math. Soc. 51(5), 1453–1467 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Liu, K., Yang, L.Z.: On entire solutions of some complex differential–difference equations. Comput. Methods Funct. Theory 13, 433–447 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Naftalevich, A.: On a differential–difference equation. Mich. Math. J. 22, 205–223 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  15. Naftalevich, A.: On meromorphic solutions of a linear differential–difference equation with constant coefficients. Mich. Math. J. 22, 205–223 (1980)

    MathSciNet  MATH  Google Scholar 

  16. Naftalevich, A.: Meromorphic solutions of a differential–difference equation. Uspekhi Mat. Nauk. 99(3), 191–196 (1961). (In Russian)

    MathSciNet  Google Scholar 

  17. Quispel, G.R.W., Capel, H.W., Sahadevan, R.: Continuous symmetries of differential-difference equations: the Kac-van Moerbeke equation and Painlevé reduction. Phys. Lett. A 170, 379–383 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, S.M., Li, S.: On entire solutions of nonlinear difference–differential equations. Bull. Korean. Math. Soc. 50(5), 1471–1479 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Wu, S.Z., Zheng, X.M.: Growth of meromorphic solutions of complex linear differential–difference equations with coefficients having the same order. J. Math. Res. Appl. 34(6), 683–695 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Yang, C.C., Yi, H.X.: Uniqueness Theory of Meromorphic Functions. Kluwer Academic Publishers, Dordrecht (2003)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kai Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, K., Song, C.J. Meromorphic Solutions of Complex Differential–Difference Equations. Results Math 72, 1759–1771 (2017). https://doi.org/10.1007/s00025-017-0736-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00025-017-0736-y

Mathematics Subject Classification

Keywords

Navigation