Let $m,k$ be positive integers with $k\le m$. Does anyone know some hypergeometric identities that imply $$\sum_{j=0}^k\frac{(1/2)_{kj}(m+1)_j(m)_j}{(1/2)_j(kj)!j!} =\frac{(m\frac{1}{2})_k(m+\frac{1}{2})_k}{(\frac{1}{2})_kk!}$$ where $a_k=a(a+1)\cdots (a+k1)$ is the Pochhammer's symbol.

Treating both sides as coefficients of $x^k$ in a power series with variable $x$, the lefthand side turns into a powerseries product, one factor representing $\sqrt{1x}$. We end up with a special case of the known hypergeometric identity $$(1x)^{a+bc} {}_2F_1(a,b;c;x) = {}_2F_1(ca,cb;c;x)$$ with $a = m+1$, $b = m$, $c = 1/2$. Cf. (34) of this MathWorld entry. 


Equivalently to ccorn's answer, the sum may be written as $$\frac {(1/2)_{k}}{k!}{}_{3}F_{2}\left({{m+1,m,k}\atop {3/2k,1/2}} \Bigm 1\right)$$ which may be evaluated by Saalschütz's theorem (also called the PfaffSaalschütz theorem). 

