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

The authors consider the linear functional equation

j=0 n a j (z)f(c j z)=Q(z)( FE )

where c{0}, n, the coefficients a 0 ,a 1 ,,a n ,Q are given complex functions, and f: is the unknown function to be determined. The authors show that if 0<|c|<1, the coefficients a 0 ,a 1 ,,a n are complex constants, Q(z) is a meromorphic function, and j=0 n a j c jk 0 for all k, 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 λ(1/f) of poles and λ(f) of zeros of F.

39B32Functional equations for complex functions
30D05Functional equations in the complex domain, iteration and composition of analytic functions
30D35Distribution of values (one complex variable); Nevanlinna theory