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


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.


Volume 42, Issue 6
November and December 2016
Pages 1497-1505
  • Receive Date: 27 April 2014
  • Revise Date: 22 September 2015
  • Accept Date: 23 September 2015