Copyright © 2010 Jian-jun Zhang and Liang-wen Liao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main purpose of this paper is to present some properties of the meromorphic solutions of complex difference equation of the form ∑λ∈Iαλ(z)(∏ν=1nf(z+cν)lλ,ν)/∑μ∈Jβμ(z)(∏ν=1nf(z+cν)mμ,ν)=R(z,f(z)), where I={λ=(lλ,1,lλ,2,…,lλ,n)∣lλ,ν∈ℕ  ∪  {0}, ν=1,2,…,n} and J={μ=(mμ,1,mμ,2,…,mμ,n)∣mμ, ν∈ℕ  ∪  {0}, ν=1,2,…,n} are two finite index sets, cν  (ν=1,2,…,n) are distinct, nonzero complex numbers, αλ(z)  (λ∈I) and βμ(z)  (μ∈J) are small functions relative to f(z),R(z,f(z)) is a rational function in f(z) with coefficients which are small functions of f(z). We also consider related complex functional equations in the paper.