On the Analytic Solution of Certain Functional and Difference Equations

Let P(u, v) be an irreducible polynomial with complex coefficients and let q >=slant 2 be an integer. We establish the necessary and sufficient conditions under which the functional equation begin{equation*}tag{F} P(f(z),f(z^q)) = 0,end{equation*} has a non-constant analytic solution that is either regular in a neighbour-hood of the point z = 0 or has a pole at this point (theorem 1). By a simple change of variable, the difference equation begin{equation*}tag{D} (F(Z),F(Z+1)) = 0,end{equation*} can be proved under the same restrictions to have a non-constant solution of the form F(Z) = sum^∞_j=if_je^{-jq^Z}, which is regular in the strip Re Z >=slant X_0, |Im Z| < π/2 ln q, if X_0 is sufficiently large (theorem 2).