Finite order meromorphic solutions of linear difference equations

Abstract

In this paper, we mainly investigate the growth and the value distribution of meromorphic solutions of the linear difference equation \begin{equation*} a_{n}(z)f(z+n)+…+a_{1}(z)f(z+1)+a_{0}(z)f(z)=b(z), \end{equation*} where $a_{0}(z),a_{1}(z),\cdots,a_{n}(z),b(z)$ are entire functions such that $a_{0}(z)a_{n}(z)\not\equiv 0$. For a finite order meromorphic solution $f(z)$, some interesting results on the relation between $\rho=\rho(f)$ and $\lambda_{f}=\max\{\lambda(f),\lambda(1/f)\}$, are proved. And examples are provided for our results.