Finding closed form for a product

Question

I have no idea how to solve the below product whose closed form I need to solve a problem, can anyone at the very least guide me to a solution or give me a source to check?

$\prod_{k=1}^{n}\dfrac{2^k-1}{2^k}$

Answer 1

The closest thing to a closed form I can find is in terms of the q-Pochhammer symbol,

$$\begin{align*} \prod_{k=1}^n\frac{2^k-1}{2^k} &=\frac{1}{2^{n(n+1)/2}}\prod_{k=1}^n\left(2^k-1 \right ) \\ &=\frac{(-1)^n}{2^{n(n+1)/2}}\prod_{k=1}^n\left(1-2^k \right ) \\ &=\frac{(-1)^n}{2^{n(n+1)/2}}\prod_{k=0}^{n-1}\left(1-2^{k+1} \right ) \\ &=\frac{(-1)^n}{2^{n(n+1)/2}}(2;\,2)_n. \end{align*}$$

Or this may be written as

$$\prod_{k=1}^n\frac{2^k-1}{2^k}=\left(\frac{1}{2};\,\frac{1}{2}\right)_n.$$