Product of r Choose k with r Minus Half Choose k/Formulation 2

Theorem
Let $k \in \Z$, $r \in \R$.


 * $\dbinom r k \dbinom {r - \frac 1 2} k = \frac {\dbinom {2 r} {2 k} \dbinom {2 k} k} {4^k}$

where $\dbinom r k$ denotes a binomial coefficient.

Proof
From Binomial Coefficient expressed using Beta Function:


 * $(1): \quad \dbinom r k \dbinom {r - \frac 1 2} k = \dfrac 1 {\left({r + 1}\right) \Beta \left({k + 1, r - k + 1}\right) \left({r + \frac 1 2}\right) \Beta \left({k + 1, r - k + \frac 1 2}\right)}$

Then:

Then: