I am trying to find out where there are closed-form for products of the form
$$ \prod_{n=0}^{\infty} (1+ f_n(x) ) $$
In particular, to make it simpler, Im trying to find out for $f_n (x) = -x^{2^n} $. In other words, we are trying to find closed form for $\prod_n (1-x^{2^n}) $.
I know that if we had $1+x^{2^n}$ instead, then because of $1+x^{2^n} = \dfrac{1-x^{2^{n+1} }}{1-x^{2^n} } $, then that product have an easy closed form. But, I am stuck in trying to find it for $\prod_n (1-x^{2^n}) $. Is it possible?