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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 66, Issue 3


Meromorphic solutions of q-shift difference equations

Xiaoguang Qi / Lianzhong Yang / Yong Liu
Published Online: 2016-08-23 | DOI: https://doi.org/10.1515/ms-2015-0168


In this paper, we consider the growth of meromorphic solutions of some generalized q-shift difference equation An(z)f(qz + n) + … + A1(z)f(qz + 1) + A0(z)f(qz) = 0. Moreover, we also investigate the existence of transcendental meromorphic solutions of Fermat type of q-shift difference equations.

MSC 2010: Primary 39A05; Secondary 30D35

Keywords: meromorphic functions; growth; q-shift difference equation

This work was supported by the National Natural Science Foundation of China (No. 11301220 and No. 11371225) and the Tianyuan Fund for Mathematics (No. 11226094), the NSF of Shandong Province, China (No. ZR2012AQ020 and No. ZR2010AM030) and the Fund of Doctoral Program Research of University of Jinan (XBS1211)


  • [1]

    Batchelder, P. M.: An Introduction to Linear Difference Equations, Dover Publications, Inc., New York, 1967.Google Scholar

  • [2]

    Bergweiler, W.—Ishizaki, K.—Yanagihara, N.: Meromorphic solutions of some functional equations, Methods Appl. Anal. 6 (1999), 617–618. Correction: Methods Appl. Anal. 6 (1999), 617–618.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 (2008), 105–129.Google Scholar

  • [4]

    Chen, Z. X.: On growth, zeros and poles of meromorphic solutions of linear and nonlinear difference equations, Sci. China Math. 54 (2011), 2123–2133.Google Scholar

  • [5]

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

  • [6]

    Halburd, R. G.—Korhonen, R. J.: Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. 31 (2006), 463–478.Google Scholar

  • [7]

    Hayman, W.: Meromorphic Functions, Clarendon Press, Oxford, 1964.Google Scholar

  • [8]

    Heittokangas, J.—Korhonen, R.—Laine, I.—Rieppo, J.—Zhang, J. L.: Value sharing results for shifts of meromorphic functions, and sufficient conditions for periodicity, J. Math. Anal. Appl. 355 (2009), 352–363.Google Scholar

  • [9]

    Gross, F.: On the equation fn + gn = 1, Bull. Amer. Math. Soc. 72 (1966), 86–88.Google Scholar

  • [10]

    Gross, F.: On the equation fn + gn = hn, Amer. Math. Monthly 73 (1966), 1093–1096.Google Scholar

  • [11]

    Korhonen, R.: An extension of Picards theorem for meromorphic functions of small hyper-order, J.Math. Anal. Appl. 357 (2009), 244–253.Google Scholar

  • [12]

    Li, S.—GAO, Z. S.: Finite order meromorphic solutions of linear difference equations, Proc. Japan Acad. 87 (2011), 73–76.Google Scholar

  • [13]

    Liu, K.—Qi, X. G.: Meromorphic solutions of q-shift difference equations, Ann. Polon. Math. 101 (2011), 215–225.Google Scholar

  • [14]

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

  • [15]

    Nörlund, N. E.: Differenzenrechnung, Springer, Berlin, 1924 (German).Google Scholar

  • [16]

    Qi, X. G.—Liu, Y.—Yang, L. Z.: Nevanlinna theory for the f(qz + c) and its applications, Acta Math. Sci. Ser. A Chin. Ed. 33 (2013), 819–828.Google Scholar

  • [17]

    Shimomura, S.: Entire solutions of a polynomial difference equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), 253–266.Google Scholar

  • [18]

    Whittaker, J. M.: Interpolatory Function Theory, Cambridge University Press/Macmillan, New York, 1935.Google Scholar

  • [19]

    Yanagihara, N.: Meromorphic solutions of some difference equations, Funkcial. Ekvac. 23 (1980), 309–326.Google Scholar

  • [20]

    Yanagihara, N.: Meromorphic solutions of some difference equations II, Funkcial. Ekvac. 24 (1981), 113–124.Google Scholar

  • [21]

    Yanagihara, N.: Meromorphic solutions of some difference equations of nth order, Arch. Ration. Mech. Anal. 91 (1985), 169–192.Google Scholar

  • [22]

    Yang, C. C.—Yi, H. X.: Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers, Dordrecht-Boston-London, 2003.Google Scholar

  • [23]

    Zhang, J. L.—Korhonen, R.: On the Nevanlinna characteristic of f(qz) and its applications, J. Math. Anal. Appl. 369 (2010), 537–544.Google Scholar

About the article

Received: 2013-04-30

Accepted: 2013-09-02

Published Online: 2016-08-23

Published in Print: 2016-06-01

Citation Information: Mathematica Slovaca, Volume 66, Issue 3, Pages 667–676, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2015-0168.

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