Some results on value distribution of the difference operator

In this article, we consider the uniqueness of the difference monomials f n (z)f(z+c). Suppose that f(z) and g(z) are transcendental meromorphic functions with finite order and E k (1,f n (z)f(z+c))=E k (1,g n (z)g(z+c)). Then we prove that if one of the following holds (i) n≥14 and k≥3, (ii) n≥16 and k=2, (iii) n≥22 and k=1, then f(z)≡t 1 g(z) or f(z)g(z)=t 2, for some constants t 1 and t 2 that satisfy t n+1 1 =1 and t n+1 2 =1. We generalize some previous results of Qi et. al.