Some results on value distribution of the difference operator

Abstract

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) $n geq 14$ and $kgeq 3$, (ii) $n geq 16$ and $k=2$, (iii) $n geq 22$ and $k=1$, then $f(z)equiv t_1g(z)$ or $f(z)g(z)=t_2,$ for some constants $t_1$ and $t_2$ that satisfy $t_1^{n+1}=1$ and $t_2^{n+1}=1$. We generalize some previous results of Qi et. al.