## Abstract

Let $P(u, v)$ be an irreducible polynomial with complex coefficients and let $q \geqslant 2$ 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 $F(Z) = \sum^\infty_{j=i}f_je^{-jq^Z},$ which is regular in the strip $\mathrm{Re} Z \geqslant X_0, |\mathrm{Im} Z| < \pi/2 \ln q,$ if X$_0$ is sufficiently large (theorem 2).