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

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


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

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

Proof
From Binomial Coefficient expressed using Beta Function:


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

Then:

Then: