Binomial Coefficient of Minus Half

Theorem

Let $k \in \Z$.

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

where $\dbinom {-\frac 1 2} k$ denotes a binomial coefficient.

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

Setting $r = -\dfrac 1 2$:

$\ds \dbinom {-\frac 1 2} k \dbinom {-1} k$

$=$

$\ds \frac 1 {4^k} \dbinom {-1} k \dbinom {-1 - k} k$

$\ds \leadsto \ \ $

$\ds \dbinom {-\frac 1 2} k$

$=$

$\ds \frac 1 {4^k} \dbinom {-1 - k} k$

$\ds $

$=$

$\ds \frac {\paren {-1}^k} {4^k} \dbinom {k - \paren {-1 - k} - 1} k$

$\ds $

$=$

$\ds \frac {\paren {-1}^k} {4^k} \dbinom {2 k} k$

simplifying

$\blacksquare$