The following is from chapter 1, problem 10 from Wilf's Generatingfunctionology. The first part of the problem states that you are given a function defined for $n\geq 1$ with the following relations, $f(1)=1,\ f(2n)=f(n),\ f(2n+1)=f(n)+f(n+1)$, with $$F(x)=\sum\limits_{n\geq 1}f(n)x^{n-1}$$
and show that this satisfies the relation, $F(x)=(1+x+x^2)F(x^2)$. I have done this, it is the second part of the question I'm concerned about as it's slightly outside of the content covered in the chapter. It asks to prove that, $$F(x)=\prod\limits_{j\geq 0}^{\infty}\left\{1+x^{2^j}+x^{2^{j+1}}\right\}$$ Obviously, this is true for finite $n$ as we can inductively just keep applying the relation, ie $F(x^2)=(1+x^2+x^4)F(x^4)$, so plugging into the original gives, $F(x)=(1+x+x^2)(1+x^2+x^4)F(x^4)$ and so on, but I'm sort of at a loss how to prove this infinite product in general, or just a technique to do so. This book has solutions, but the explanation for this part is not given and just stated as "obvious", which it is not to me. Any help appreciated.
Two parts:
Does the given product satisfy the relationship?
Is it a unique solution? I.e. can there be more than one function which satisfies the relationship and also evaluates to $f(1)$ at $x=0$? Suppose $F(x) = (1 + x+ x^2)F(x^2)$, $G(x) = (1 + x + x^2)G(x^2)$, $F(0) = G(0) = 1$. Then $F(x) - G(x) = (1 + x + x^2)\left(F(x^2) - G(x^2)\right)$ and you can do the same expansion. Then suppose that the first coefficient of $x^a$ which differs is finite, and derive a contradiction by expanding $k + \lg a$ times.