On growth of meromorphic solutions of complex functional difference equations (Q1725080)

Summary: The main purpose of this paper is to investigate the growth order of the meromorphic solutions of complex functional difference equation of the form \((\sum_{\lambda \in I} \alpha_\lambda(z)(\prod_{\nu = 1}^n f(z + c_\nu)^{l_{\lambda, \nu}})) /(\sum_{\mu \in J} \beta_\mu(z)(\prod_{\nu = 1}^n f(z +c_\nu)^{m_{\mu, \nu}})) =Q(z, f(p(z)))\), where \(I =\{\lambda = (l_{\lambda, 1}, l_{\lambda, 2}, \ldots, l_{\lambda, n}) \mid l_{\lambda, \nu} \in \mathbb N \bigcup \{0 \},\;\nu =1,2, \ldots, n \}\) and \(J = \{\mu = (m_{\mu, 1}, m_{\mu, 2}, \ldots, m_{\mu, n}) \mid m_{\mu, \nu} \in \mathbb N \bigcup \{0 \}, \;\nu = 1,2, \ldots, n \}\) are two finite index sets, \(c_\nu(\nu = 1,2, \ldots, n)\) 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 \(f(z)\), \(p(z) = p_k z^k + p_{k - 1} z^{k - 1} + \cdots + p_0 \in \mathbb C [z]\) of degree \(k \geq 1\). We also give some examples to show that our results are sharp.