Find a general formula for the following product: $$S_{n} = \prod_{k=1}^{n}(1-x^{2^{k-1}})$$
I know how to evaluate the following product:
$$S_{n} = \prod_{k=1}^{n}(1+x^{2^{k-1}})$$ by multiplying both sides by $1-x$ everything simplifies and we get $\frac{1-x^{2^n}}{1-x}$ as the general formula for $S_n$.
What I was wondering is if we can find a general formula when we have an $'-'$ instead of $'+'$ sign. I tried applying the same trick, but it obviously doesn't work. I would appreciate if someone could give me their opinion.