Is there a closed form for the infinite product $\prod_{n=0}^{\infty}\bigl(1+{x \over 2^n} \bigr)$

Question

It has been a long time ago since I first encountered the following infinite product : $$ \prod_{n=0}^{\infty}\bigl(1+{x \over 2^n} \bigr) = (1 + x) \bigl(1 + {x \over 2}\bigr)\bigl(1 + {x \over 4}\bigr)\bigl(1 + {x \over 8}\bigr) \cdots $$ to my knowledge I haven't seen a closed form, but I would appreciate if gathering some information about this product. Does it relate to any known functions?

Answer 1

$$\sum_{n\geq 0}\log\left(1+\frac{x}{2^n}\right) = \sum_{n\geq 0}\sum_{m\geq 1}\frac{(-1)^{m+1}}{m}\left(\frac{x^m}{2^{nm}}\right)=\sum_{m\geq 1}\frac{(-1)^{m+1}}{m\left(1-2^{-m}\right)}x^m$$ hence the product can be written as $\exp\left(\sum_{m\geq 1}\frac{(-1)^{m+1}}{m\left(1-2^{-m}\right)}x^m\right)$, but that is far from being nice.