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

TL;DR

The paper introduces a method using composita to solve iterative functional equations of the form $A^{2^n}(x)=F(x)$, proving integer coefficient properties of solutions when $F(x)$ has integer coefficients.

Contribution

It develops a novel approach based on composita for solving specific iterative functional equations and establishes conditions for integer coefficients in solutions.

Findings

Method for solving $A^{2^n}(x)=F(x)$ using composita.

Proof that solutions have integer coefficients if $F(x)$ does.

Application of the method to generating functions with integer coefficients.

Abstract

Using the notion of the composita, we obtain a method of solving iterative functional equations of the form $A^{2^n}(x)=F(x)$, where $F(x)=\sum_{n>0} f(n)x^n$, $f(1)\neq 0$. We prove that if $F(x)=\sum_{n>0} f(n)x^n$ has integer coefficients, then the generating function $A(x)=\sum_{n>0}a(n)x^n$, 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.