Further extending results of some classes of complex difference and functional equations
The authors study a class of complex difference equations. Among other results they prove the following theorem: Let \(c_1 , c_2 , \dots , c_n\) be distinct, nonzero complex numbers and suppose that \(f(z)\) is a transcendental meromorphic solution of the difference equation \[ {{\sum_{\lambda \in I} {\alpha}_{\lambda} (z) \left ( \, \prod_{\nu =1}^n f(z+c_{\nu} )^{l_{\lambda, \nu}} \, \right)} \over {\sum_{\mu \in J} {\beta}_{\mu} (z) \left ( \, \prod_{\nu =1}^n f(z+c_{\nu} )^{m_{\mu, \nu}} \, \right )}} = R( z, f(z)) = {{a_0 (z) + a_1 (z) f(z) + \cdot + a_p (z) f(z)^p} \over {b_0 (z) + b_1 (z) f(z) + \cdot + b_p (z) f(z)^q}}, \] with coefficients \(\alpha_{\lambda}(z)\) (\(\lambda \in I\)), \(\beta_{\mu} (z)\) (\(\mu \in J\)), \(a_i(z)\) \((i=0,1,\dots , p)\), and \(b_j (z)\) (\(j=0,1, \dots , q)\) are small functions relative to \(f(z)\), where \(I = \{ \lambda = ( l_{\lambda , 1}, l_{\lambda , 2}, \dots , l_{\lambda , n})\mid l_{\lambda , \nu} \in \mathbb{N} \cup \{0\}\), \(\nu =1,2,\dots , n\}\) and \(J = \{ \mu = ( m_{\mu , 1}, m_{\mu , 2}, \dots , m_{\mu , n} )\mid m_{\mu , \nu} \in \mathbb{N} \cup \{0\}\), \(\nu =1,2,\dots , n\}\) are two finite index sets, denote \[ \sigma_{\nu} = \max_{\lambda , \mu} \{ l_{\lambda , \nu} , m_{\mu , \nu} \} \quad (\nu = 1,2, \dots , n ), \quad \sigma = \sum_{\nu=1}^n \sigma_{\nu}. \] If the order \(\rho (f) := \rho\) is finite, then \(d=\max \{ p, q\} \leq \sigma \).
complex difference and functional equations
scientific article; zbMATH DE number 5825827