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

The authors consider the linear functional equation

$\sum _{j=0}^{n}{a}_{j}\left(z\right)f\left({c}^{j}z\right)=Q\left(z\right)\phantom{\rule{2.em}{0ex}}\left(\mathrm{FE}\right)$

where $c\in ℂ\setminus \left\{0\right\}$, $n\in ℕ$, the coefficients ${a}_{0},{a}_{1},\cdots ,{a}_{n},Q$ are given complex functions, and $f:ℂ\to ℂ$ is the unknown function to be determined. The authors show that if $0<|c|<1$, the coefficients ${a}_{0},{a}_{1},\cdots ,{a}_{n}$ are complex constants, $Q\left(z\right)$ is a meromorphic function, and ${\sum }_{j=0}^{n}{a}_{j}{c}^{jk}\ne 0$ for all $k\in ℤ$, 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 $\lambda \left(1/f\right)$ of poles and $\lambda \left(f\right)$ of zeros of $F$.

MSC:
 39B32 Functional equations for complex functions 30D05 Functional equations in the complex domain, iteration and composition of analytic functions 30D35 Distribution of values (one complex variable); Nevanlinna theory