Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$

Abstract: Using the notion of the composita, we obtain a method of solving iterative functional equations of the form $A<sup>{2<sup>n}(x)=F(x)$, where $F(x)=\sum_{n&gt;0} f(n)x<sup>n$, $f(1)\neq 0$. We prove that if $F(x)=\sum_{n&gt;0} f(n)x<sup>n$ has integer coefficients, then the generating function $A(x)=\sum_{n&gt;0}a(n)x<sup>n$, which is obtained from the iterative functional equation $4A(A(x))=F(4x)$, has integer coefficients. Key words: iterative functional equation, composition of generating functions, composita.