Zong Sheng Gao, Xiao Ming Wang Beihang University, Beihang University

Abstract : In this paper, we investigate the transcendental meromorphic solutions with finite order of two different types of difference equations \begin{align*} \sum_{j=1}^{n}a_jf(z+c_j)=\frac{P(z,f)}{Q(z,f)}=\frac{\sum_{k=0}^{p}b_kf^k}{\sum_{l=0}^{q}d_lf^l} \end{align*} and \begin{align*} \prod_{j=1}^{n}f(z+c_j)=\frac{P(z,f)}{Q(z,f)}=\frac{\sum_{k=0}^{p}b_kf^k}{\sum_{l=0}^{q}d_lf^l} \end{align*} that share three distinct values with another meromorphic function. Here $a_j$, $b_k$, $d_l$ are small functions of $f$ and $a_j\not\equiv 0(j=1,2,\ldots,n)$, $b_p\not\equiv 0$, $d_q\not\equiv 0$. $c_j\neq 0$ are pairwise distinct constants. $p$, $q$, $n$ are non-negative integers. $P(z,f)$ and $Q(z,f)$ are two mutually prime polynomials in $f$.