Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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A NOTE ON MEROMORPHIC SOLUTIONS OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS

  • Received : 2018.03.22
  • Accepted : 2018.07.05
  • Published : 2019.05.31
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Abstract

In this article, we consider properties of transcendental meromorphic solutions of the complex differential-difference equation $$P_n(z)f^{(n)}(2+{\eta}_n)+{\cdots}+P_1(z)f^{\prime}(z+{\eta}_1)+P_0(z)f(z+{\eta}_0)=0$$, and its non-homogeneous equation. Our results extend earlier results by Liu et al. [9].

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References

  1. Y.-M. Chiang and S.-J. Feng, On the Nevanlinna characteristic of f($z+{\eta}$) and difference equations in the complex plane, Ramanujan J. 16 (2008), no. 1, 105-129. https://doi.org/10.1007/s11139-007-9101-1
  2. Z. X. Chen, Complex Differences and Difference Equations, Science Press, 2014.
  3. Z.-X. Chen and C.-C. Yang, On entire solutions of certain type of differential-difference equations, Taiwanese J. Math. 18 (2014), no. 3, 677-685. https://doi.org/10.11650/tjm.18.2014.3745
  4. G. G. Gundersen, Estimates for the logarithmic derivative of a meromorphic function, plus similar estimates, J. London Math. Soc. (2) 37 (1988), no. 1, 88-104. https://doi.org/10.1112/jlms/s2-37.121.88
  5. G. G. Gundersen, Finite order solutions of second order linear differential equations, Trans. Amer. Math. Soc. 305 (1988), no. 1, 415-429. https://doi.org/10.1090/S0002-9947-1988-0920167-5
  6. R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equations, J. Math. Anal. Appl. 314 (2006), no. 2, 477-487. https://doi.org/10.1016/j.jmaa.2005.04.010
  7. R. G. Halburd and R. J. Korhonen, Growth of meromorphic solutions of delay differential equations, Proc. Amer. Math. Soc. 145 (2017), no. 6, 2513-2526. https://doi.org/10.1090/proc/13559
  8. K. Liu, T. Cao, and H. Cao, Entire solutions of Fermat type differential-difference equations, Arch. Math. (Basel) 99 (2012), no. 2, 147-155. https://doi.org/10.1007/s00013-012-0408-9
  9. K. Liu and C. J. Song, Meromorphic solutions of complex differential-difference equations, Results Math. 72 (2017), no. 4, 1759-1771. https://doi.org/10.1007/s00025-017-0736-y
  10. K. Liu and L. Yang, On entire solutions of some differential-difference equations, Comput. Methods Funct. Theory 13 (2013), no. 3, 433-447. https://doi.org/10.1007/s40315-013-0030-2
  11. A. Naftalevich, Meromorphic solutions of a differential-difference equation, Uspehi Mat. Nauk 16 (1961), no. 3 (99), 191-196.
  12. A. Naftalevich, On a differential-difference equation, Michigan Math. J. 22 (1975), no. 3, 205-223 (1976).
  13. X. Qi and L. Yang, Properties of meromorphic solutions to certain differential-difference equations, Electron. J. Differential Equations 2013, No. 135, 9 pp.
  14. S. Shimomura, Entire solutions of a polynomial difference equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 2, 253-266.
  15. S. Wu and X. Zheng, Growth of meromorphic solutions of complex linear differential-difference equations with coefficients having the same order, J. Math. Res. Appl. 34 (2014), no. 6, 683-695.
  16. N. Yanagihara, Meromorphic solutions of some difference equations, Funkcial. Ekvac. 23 (1980), no. 3, 309-326.
  17. N. Yanagihara, Meromorphic solutions of some difference equations of nth order, Arch. Rational Mech. Anal. 91 (1985), no. 2, 169-192. https://doi.org/10.1007/BF00276862
  18. C.-C. Yang and H.-X. Yi, Uniqueness Theory of Meromorphic Functions, Mathematics and its Applications, 557, Kluwer Academic Publishers Group, Dordrecht, 2003.