Finding the closed form of $\prod _{n=0}^{\infty }\left(1\:-\:x^{2^n}+x^{2^{n+1}}\right)$

Question

As far as I can tell this expression doesn't telescope. How would one go about this?

Answer 1

Note that
$$(1 - x^{2^n} + x^{2^{n+1}})(1+x^{2^n}) = 1 + (x^{3})^{2^n}\implies \prod_{n\geq 0}(1 - x^{2^n} + x^{2^{n+1}}) = \frac{f(x^3)}{f(x)}$$ where $f(x) = \prod_{n\geq 0}(1+x^{2^n})$ so the problem reduces to evaluating $f(x)$. For this you can use $$(1-x^{2^n})(1+x^{2^n}) = (1-x^{2^{n+1}}) \implies (1+x^{2^n}) = \frac{(1-x^{2^{n+1}})}{(1-x^{2^n})}$$ which we see will telescope when you take the product.

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Another way is to notice that the partial products of the original series when multiplied with $(1+x+x^2)$ gives us $$(1-x+x^2)\cdot (1+x+x^2) = 1 + x^2 + x^4$$ $$(1-x+x^2)(1-x^2+x^4)\cdot (1+x+x^2) = 1 + x^4 + x^8$$ $$(1-x+x^2)(1-x^2+x^4)(1-x^4+x^8)\cdot (1+x+x^2) = 1 + x^8 + x^{16}$$ which shows a pattern for the partial products that you can prove using induction.