I was wondering about the function $$F(x) = \prod_{n=0}^{\infty}{(1+x^{4^{n}})} = 1+x+x^4+x^5+x^{16}+x^{17}+...$$
where the exponents in the resulting power series are given by the Moser-de Bruijn sequence.
Wikipedia says that it has the functional equations $$F(x)F(x^2)=\frac{1}{1-x}$$ and $$F(x)=(1+x)F(x^4).$$
My question is: Is there an analytic continuation of this function outside of its radius of convergence? Is it possible to represent this series using a definite integral? What other identities does this function have?
The product represention suggests that the function should have singularities at every $4^n$th root of $1$, since all terms in the product after a certain point would equal $2$. Also, the Hadamard gap theorem says that if the asymptomatic ratio between non-zero exponents is greater than 1, the function can't be analytically continued, while the Moser-de Bruijn sequence is proportional to the square numbers.