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Meromorphic solutions of certain difference equations of first order

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Abstract

In this article, we investigate the growth and value distribution of meromorphic solutions of a first order difference equation with small coefficients in the complex plane.

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References

  1. Bergweiler W., Langley J.K.: Zeros of differences of meromorphic functions. Math. Proc. Camb. Philos. Soc. 142, 133–147 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Clunie J.: On integral and meromorphic functions. J. Lond. Math. Soc. 37, 17–27 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chiang Y.M., Feng S.J.: On the Nevanlinna characteristic of f(z +  η) and difference equations in the complex plane. Ramanujan J. 16, 105–129 (2009)

    Article  MathSciNet  Google Scholar 

  4. Chiang Y.M., Feng S.J.: On the growth of logarithmic differences. difference quotients and logarithmic derivatives of meromorphic functions. Trans. Am. Math. Soc. 361, 3767–3791 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Chen Z.X.: The zero, pole and order of meromorphic solutions of differential equations with meromorphic coefficients. Kodai Math. J. 19, 341–354 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gundersen G.G.: Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates. J. Lond. Math. Soc. 37, 88–104 (1988)

    MATH  MathSciNet  Google Scholar 

  7. Halburd R.G., Korhonen R.: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math. 94, 463–478 (2006)

    MathSciNet  Google Scholar 

  8. Halburd R.G., Korhonen R.: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Appl. 314, 477–487 (2006)

    MATH  MathSciNet  Google Scholar 

  9. Halburd R.G., Korhonen R.: Finite order solutions and the discrete Painlevé equations. Proc. Lond. Math. Soc. 94, 443–474 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  11. Hayman W.K.: The local growth of power series: a survey of the Wiman-Valiron method. Canad. Math. Bull. 17, 317–358 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ishizaki, K.: Meromorphic solutions of complex differential equations. Doctoral Dissertation, Chiba (1993)

  13. Ishizaki K., Yanagihara N.: Wiman-Valiron method for difference equations. Nagoya Math. J. 175, 75–102 (2004)

    MATH  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  15. Laine I., Yang C.C.: Clunie theorems for difference and q-difference polynomials. J. Lond. Math. Soc. 76(2), 556–566 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Mohon’ko, A.Z.: The Nevanlinna characteristics of certain meromorphic functions, Teor. Funktsii Funktsional. Anal. i Prilozhen 14, 83–87 (1971). (Russian)

  17. Mohon’ko, A.Z., Mohon’ko, V.D.: Estimates of the Nevanlinna characteristics of certain classes of meromorphic functions, and their applications to differential equations. Sibirsk. Mat. Zh. 15, 1305–1322 (1974). (Russian)

    Google Scholar 

  18. Steinmetz N.: Ein Malmquistscher Satz für algebraische Differentialgleichungen erster Ordnung. J. Reine Angew. Math. 316, 44–53 (1980)

    MATH  MathSciNet  Google Scholar 

  19. Yang C.C., Yi H.X.: Uniqueness of Meromorphic Functions. Kluwer, Dordrecht (2003)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shaoxing College of Arts and Sciences, Shaoxing, 312000, Zhejiang, People’s Republic of China

    Yong Liu

  2. Department of Physics and Mathematics, University of Eastern Finland, Joensuu Campus, P.O. Box 111, 80101, Joensuu, Finland

    Yong Liu

Authors
  1. Yong Liu
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Correspondence to Yong Liu.

Additional information

The work was supported by the NNSF of China (No.10771121), the NSF of Shangdong Province, China (No Z2008A01), Shandong university graduate student independent innovation fund(yzc11024) and CIMO.

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Liu, Y. Meromorphic solutions of certain difference equations of first order. Aequat. Math. 87, 309–323 (2014). https://doi.org/10.1007/s00010-013-0192-z

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  • DOI: https://doi.org/10.1007/s00010-013-0192-z

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