Meromorphic solutions of some linear functional equations (Q1587847)

The authors consider the linear functional equation \[ \sum_{j=0}^n a_j(z) f( c^j z)= Q(z) \tag{FE} \] where \(c \in \mathbb{C} \setminus \{ 0\}\), \(n \in \mathbb{N}\), the coefficients \(a_0, a_1, \dots ,a_n, Q\) are given complex functions, and \(f : \mathbb{C} \to \mathbb{C}\) is the unknown function to be determined. The authors show that if \(0< |c|< 1\), the coefficients \(a_0, a_1, \dots ,a_n\) are complex constants, \(Q(z)\) is a meromorphic function, and \(\sum_{j=0}^n a_j c^{jk} \neq 0\) for all \(k \in \mathbb{Z}\), then exactly one meromorphic solution of the functional equation (FE) exists. In the general case, the authors give growth estimates for the solution \(f\) as well as the exponent of convergence \(\lambda (1/f)\) of poles and \(\lambda (f)\) of zeros of \(F\).