Meromorphic solutions of generalized Schröder equations

Summary.

We consider meromorphic solutions of functional equations of the form¶¶$f(cz) = R(z,f(z)) = {\sum_{j=0}^pa_j(z)f(z)^j \over \sum_{j=0}^q b_j(z)f(z)^j}$,¶where the coefficients a _j (z),b _j (z) are meromorphic functions and c is a complex constant. In fact, for $|c| > 1$, any local meromorphic solution around the origin has a meromorphic continuation over ${\Bbb C}$. We prove a number of results on the growth and value distribution of solutions. In the special case of¶¶$f(cz) = A(z) + \gamma f(z) + \delta f(z)^2$,¶where $c, \gamma, \delta \in {\Bbb C}$, $|c|>1$, $\delta \neq 0$, and A(z) is entire, we offer a detailed analysis on the number of distinct meromorphic solutions.