Value Distribution of Certain Type of Difference Polynomials

Abstract

We investigate the value distribution of difference product $f(z{)}^n{\sum }_{i=\mathrm{1}}^k\mathrm{‍}{a}_{i}f(z+c_i)$, for $n\ge \mathrm{2}$ and $n=\mathrm{1}$, respectively, where $f(z)$ is a transcendental entire function of finite order and $a_i,\mathrm{}{c}_i$ are constants satisfying ${\sum }_{i=\mathrm{1}}^k\mathrm{‍}{a}_{i}f(z+c_i)\not\equiv \mathrm{0}$.