Meromorphic solutions of some linear functional equations. (English) Zbl 0963.39028

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 \(F\).


39B32 Functional equations for complex functions
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
Full Text: DOI