Let $m,k$ be positive integers with $k\le m$. Does anyone know some hypergeometric identities that imply 
$$\sum\_{j=0}^k\frac{(-1/2)\_{k-j}(m+1)\_j(-m)\_j}{(1/2)\_j(k-j)!j!}
=\frac{(-m-\frac{1}{2})\_k(m+\frac{1}{2})\_k}{(\frac{1}{2})\_kk!}$$ 
where $a\_k=a(a+1)\cdots (a+k-1)$ is the Pochhammer's symbol.



Hypergeometric identities
Treating both sides as coefficients of $x^k$ in a power series with variable $x$, the left-hand side turns into a power-series product, one factor representing $\sqrt{1-x}$. We end up with a special case of the known hypergeometric identity
$$(1-x)^{a+b-c} {}\_2F\_1(a,b;c;x) = {}\_2F\_1(c-a,c-b;c;x)$$
with $a = m+1$, $b = -m$, $c = 1/2$.
Cf. (34) of [this MathWorld entry](http://mathworld.wolfram.com/HypergeometricFunction.html).


