Growth of meromorphic solutions for complex difference‎ ‎equations of Malmquist type

Document Type : Research Paper

Authors

LMIB & School of Mathematics and Systems Science‎, ‎Beihang University‎, ‎Beijing‎, ‎100191‎, ‎China.

Abstract

‎In this paper‎, ‎we give some necessary conditions for a complex‎ ‎difference equation of Malmquist type‎
$$‎\sum^n_{j=1}f(z+c_j)=\frac{P(f(z))}{Q(f(z))}‎,$$
‎where $n(\in{\mathbb{N}})\geq{2}$‎, ‎and $P(f(z))$ and $Q(f(z))$ are‎ ‎relatively prime polynomials in $f(z)$ with small functions as‎ ‎coefficients‎, ‎admitting a meromorphic function of finite order‎. ‎Moreover‎, ‎the properties of finite order transcendental meromorphic‎ ‎solutions for complex difference equation‎ ‎$\prod^n_{j=1}f(z+c_j)=P(f(z))/Q(f(z))$ are also investigated.

Keywords

Main Subjects

References

M. J. Ablowitz, R. Halburd and B. Herbst, On the extension of Painleve property to difference equations, Nonlinearity 13 (2000), no. 3, 889--905.
Z. X. Chen and K. H. Shon, Value distribution of meromorphic solutions of certain difference Painleve equations, J. Math. Anal. Appl. 364 (2010), no. 2, 556--566.
Y. M. Chiang and S.-J. Feng, On the Nevanlinna charactericstic of f(z+µ) and difference equations in the complex plane, Ramanujan J. 16 (2008), no. 1, 105--129.
W. K. Hayman, Meromorphic Functions, Clarendon Press, Oxford, 1964.
J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo and K. Tohge, Complex difference equations of Malmquist type, Comput. Methods Funct. Theory 1 (2001), no. 1, 27--39.
R. G. Halburd and R. J. Korhonen, Finite order solutions and the discrete Painleve
equations, Proc. Lond. Math. Soc. 94 (2007), no. 3, 443--474.
R. G. Halburd and R. J. Korhonen, Difference analogue of the lemma on the logarithmic derivative with applications to difference equation, J. Math. Anal. Appl. 314 (2006), no. 2, 477--487.
Z. B. Huang and Z. X. Chen, On properties of meromorphic solutions for complex difference equations of Malmquist type, Acta Math. Sci. Ser. B 33 (2013), no. 4, 1141--1152.
I. Laine and C. C. Yang, Clunie theorems for difference and q-difference polynomials, J. Lond. Math. Soc. 76 (2007), no. 3, 556--566.
I. Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, Berlin, 1993.
J. L. Zhang and R. Korhonen, On the Nevanlinna characteristic of f(qz) and its applications, J. Math. Anal. Appl. 369 (2010), no. 2, 537--544.
Bulletin of the Iranian Mathematical Society
Volume 42, Issue 6
December 2016
Pages 1497-1505

Files

History

  • Receive Date: 27 April 2014
  • Revise Date: 22 September 2015
  • Accept Date: 23 September 2015
  • First Publish Date: 18 December 2016

Share

How to cite

Statistics