迭代函数方程$A^{2^n}(x)=F(x)$的求解方法
Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$
摘要：
利用复合的概念，得到了求解$A^{2^n}(x)=F(x)$型迭代函数方程的一种方法，其中$F(x)=\sum_{n>0} f(n)x^n$，$f(1)\neq 0$ .证明了若$F(x)=\sum_{n>0} f(n)x^n$具有整数系数，则由迭代函数方程$4A(A(x))=F(4x)$得到的生成函数$A(x)=\sum_{n>0}a(n)x^n$具有整数系数.关键词：迭代函数方程，母函数的合成，合成。
英文摘要：
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.
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