On Growth of Meromorphic Solutions of Complex Functional Difference Equations

Abstract

The main purpose of this paper is to investigate the growth order of the meromorphic solutions of complex functional difference equation of the form $({\sum }_{\lambda \in I}‍{\alpha }_{\lambda }(z)({\prod }_{\nu =1}^n‍f(z+c_{\nu }{)}^{l_{\lambda ,\nu }}))/({\sum }_{\mu \in J}‍{\beta }_{\mu }(z)({\prod }_{\nu =1}^n‍f(z+$ $c_{\nu }{)}^{m_{\mu ,\nu }}))=$ $Q(z,f(p(z)))$, where $I=$ $\{\lambda =(l_{\lambda ,1},l_{\lambda ,2},\dots ,l_{\lambda ,n})\mid l_{\lambda ,\nu }\in \mathbb{N}\bigcup ‍\{0\}\text{,}\nu =$ $\mathrm{1,2},\dots ,n\}$ and $J=\{\mu =(m_{\mu ,1},m_{\mu ,2},\dots ,m_{\mu ,n})\mid m_{\mu ,\nu }\in \mathbb{N}\bigcup ‍\{0\}\text{,}\nu =\mathrm{1,2},\dots ,n\}$ are two finite index sets, $c_{\nu }(\nu =\mathrm{1,2},\dots ,n)$ are distinct complex numbers, ${\alpha }_{\lambda }(z)(\lambda \in I)$ and ${\beta }_{\mu }(z)(\mu \in J)$ are small functions relative to $f(z),$ and $Q(z,u)$ is a rational function in $u$ with coefficients which are small functions of $f(z)$, $p(z)=p_k{z}^k+p_{k-1}{z}^{k-1}+\cdots +p_0\in ℂ[z]$ of degree $k\ge 1$. We also give some examples to show that our results are sharp.