Hello and Happy holidays.

I am interested in generalizations of the following product formula for the gamma function $\Gamma(z)= \int_{0}^{\infty} t^{z-1}e^{-t}dt$ when $n \geq 2$:

\begin{align} \displaystyle\prod_{k = 1}^{n} \frac{\Gamma(\frac{z}{2^k}+\frac{1}{2})}{\Gamma(\frac{1}{2})} = & \frac{\Gamma(z+1)}{2^{2z(1-\frac{1}{2^n})} \Gamma(\frac{z}{2^n}+\frac{1}{2})} \end{align}

Let $H_1,H_2,...H_n \in (0,1)$ and $z \in \mathbb{R^+}$.

1) Then is it true that the following formula holds for $n \geq 2$?

\begin{align} \frac{\Gamma(zH_1 + \frac{1}{2})\Gamma(zH_1H_2 + \frac{1}{2}) \dotsb \Gamma(zH_1H_2 \dotsb H_n + \frac{1}{2})}{\prod_{k=1}^{n} \Gamma(\frac{1}{2})} = \end{align}

$\frac{\Gamma(z+1)}{2^{2z(1-H_1H_2 \dotsb H_n)} \Gamma( z H_1 H_2 \dotsb H_n + \frac{1}{2})}$

2) As $n$ tends to $\infty$ is the LHS of the last expression finite?

3) Does question 1) hold if $H_1 = 1$?

(In the context of my research the $H_i$'s are Hurst parameters from n+1 independent fractional Brownian motions)