DOI QR코드

DOI QR Code

MEROMORPHIC SOLUTIONS OF SOME q-DIFFERENCE EQUATIONS

  • Chen, Baoqin ;
  • Chen, Zongxuan
  • Received : 2010.08.14
  • Published : 2011.11.30

Abstract

We consider meromorphic solutions of q-difference equations of the form $$\sum_{j=o}^{n}a_j(z)f(q^jz)=a_{n+1}(z),$$ where $a_0(z)$, ${\ldots}$, $a_{n+1}(z)$ are meromorphic functions, $a_0(z)a_n(z)$ ≢ 0 and $q{\in}\mathbb{C}$ such that 0 < |q| ${\leq}$ 1. We give a new estimate on the upper bound for the length of the gap in the power series of entire solutions for the case 0 < |q| < 1 and n = 2. Some growth estimates for meromorphic solutions are also given in the cases 0 < |q| < 1. Moreover, we investigate zeros and poles of meromorphic solutions for the case |q| = 1.

Keywords

q-difference equation;growth;type

References

  1. M. Ablowitz, R. G. Halburd, and B. Herbst, On the extension of the Painleve property to difference equations, Nonlinearity 13 (2000), no. 3, 889-905. https://doi.org/10.1088/0951-7715/13/3/321
  2. W. Bergweiler, K. Ishizaki, and N. Yanagihara, Meromorphic solutions of some functional equations, Methods Appl. Anal. 5 (1998), no. 3, 248-258.
  3. W. Bergweiler, K. Ishizaki, and N. Yanagihara, Growth of meromorphic solutions of some functional equations I, Aequationes Math. 63 (2002), no. 1-2, 140-151. https://doi.org/10.1007/s00010-002-8012-x
  4. W. Bergweiler and J. K. Langley, Zeros of differences of meromorphic functions, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 1, 133-147. https://doi.org/10.1017/S0305004106009777
  5. B. Q. Chen, Z. X. Chen, and S. Li, Properties on solutions of some q-difference equations, Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 10, 1877-1886. https://doi.org/10.1007/s10114-010-8339-5
  6. Z. X. Chen and K. H. Shon, On zeros and fixed points of differences of meromorphic functions, J. Math. Anal. Appl. 344 (2008), no. 1, 373-383. https://doi.org/10.1016/j.jmaa.2008.02.048
  7. Y. M. Chiang and S. J. Feng, On the Nevanlinna characteristic of f($z+\eta$) and difference equations in thecomplex plane, Ramanujan J. 16 (2008), no. 1, 105-129. https://doi.org/10.1007/s11139-007-9101-1
  8. G. Gundersen, J. Heittokangas, I. Laine, J. Rieppo, and D. G. Yang, Meromorphic solutions of generalized Schroder equations, Aequationes Math. 63 (2002), no. 1-2, 110-135. https://doi.org/10.1007/s00010-002-8010-z
  9. R. G. Halburd and R. 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
  10. R. G. Halburd and R. Korhonen, Nevanlinna theory for the difference operator, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 2, 463-478.
  11. W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
  12. W. K. Hayman, Angular value distribution of power series with gaps, Proc. Lond. Math. Soc. (3) 24 (1972), no. 4, 590-624. https://doi.org/10.1112/plms/s3-24.4.590
  13. J. Heittokangas, I. Laine, J. Rieppo, and D. G. Yang, Meromorphic solutions of some linear functional equations, Aequationes Math. 60 (2000), no. 1-2, 148-166. https://doi.org/10.1007/s000100050143
  14. I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993.
  15. S. Saks and A. Zygmund, Analytic Functions, Monografie Mat. (Engl. Transl.) Tom 28, Warsaw, 1952.
  16. J. Tu and C. F. Yi, On the growth of solutions of a class of higher order linear differential equations with coefficients having the same order, J. Math. Anal. Appl. 340 (2008), no.1, 487-497. https://doi.org/10.1016/j.jmaa.2007.08.041
  17. C. C. Yang and H. X. Yi, Uniqueness Theory of Meromorphic Functions, Math. Appl., vol. 557, Kluwer Academic Publishers Group. Dordrecht, 2003.
  18. L. Yang, Value Distribution Theory and New Research, Science Press, Beijing, 1982.

Cited by

  1. FINITE LOGARITHMIC ORDER SOLUTIONS OF LINEAR q-DIFFERENCE EQUATIONS vol.51, pp.1, 2014, https://doi.org/10.4134/BKMS.2014.51.1.083

Acknowledgement

Supported by : National Natural Science Foundation of China