I Need help understanding this step in simplifying a limit equation

[mody mody]

I read a paper and i can't understand how can we make the conversion (that attached in photo)
i mean how i get formula (5) from previous one

 

[Charles Link]

I think I figured this one out. The continued product of each $j$ from $1$ to $l+1$ is the $((l+1)!)^2$. The 2's in the numerator actually can be divided in the denominator. The sequence of 1*3*5*7...(squared) in the denominator takes on two forms depending whether the last term is $(2l+1)/2$ or includes the $(l+\frac{3}{2})$ term, thereby the continued product of $(2j-1)/2$ and $(2j+1)/2$ in the denominator. The $\sqrt{\pi}$ in the denominator gets squared and moved to the numerator of the other side, and the $\frac{1}{2}$ is a simple algebraic term because $\frac{1}{(2l+3)/2}=\frac{2}{2l+3}$.