This is what I have so far:
Start with the Hypergeometrical Series$$_4F_3\left[\begin{array}{c c}\frac 12n+1,n,-x,-y\\\frac 12n,x+n+1,y+n+1\end{array};-1\right]=\dfrac {\Gamma(x+n+1)\Gamma(y+n+1)}{\Gamma(n+1)\Gamma(x+y+n+1)}\tag1$$
And let $n=-x=-y=\frac 12$. Therefore, $(1)$ takes the form
$$\begin{align*}_4F_3\left[\begin{array}{c c}\frac 54,\frac 12,\frac 12,\frac 12\\\frac 14,1,1\end{array};-1\right] & =\dfrac {1}{\Gamma\left(\frac 32\right)\Gamma\left(\frac 12\right)}\\ & =\dfrac 2\pi\tag{2}\end{align*}$$
And since the general Hypergeometrical series obeys$$_pF_q\left[\begin{array}{c c}\alpha_1,\alpha_2,\ldots,\alpha_p\\\beta_1,\beta_2,\ldots,\beta_p\end{array};x\right]=\sum\limits_{k=0}^\infty\dfrac {(\alpha_1)_k(\alpha_2)_k\cdots(\alpha_p)_k}{(\beta_1)_k(\beta_2)_k\cdots(\beta_p)_k}\dfrac {x^k}{k!}\tag{3}$$
We have the LHS as$$\begin{align*}_4F_3\left[\begin{array}{c c}\frac 54,\frac 12,\frac 12,\frac 12\\\frac 14,1,1\end{array};-1\right] & =\sum\limits_{k=0}^{\infty}\dfrac {\left(\frac 54\right)_k\left(\frac 12\right)_k\left(\frac 12\right)_k\left(\frac 21\right)_k(-1)^k}{\left(\frac 14\right)_k\left(1\right)_k\left(1\right)_kk!}\\ & =1-5\left(\dfrac 12\right)^3+9\left(\dfrac {1\cdot3}{2\cdot4}\right)^3-13\left(\dfrac {1\cdot3\cdot5}{2\cdot4\cdot6}\right)^3+\&\text c.\tag{4}\end{align*}$$
Therefore, the identity is established. However, now the question simplifies into
Question: How do we prove$$_4F_3\left[\begin{array}{c c}\frac 12n+1,n,-x,-y\\\frac 12n,x+n+1,y+n+1\end{array};-1\right]=\dfrac {\Gamma(x+n+1)\Gamma(y+n+1)}{\Gamma(n+1)\Gamma(x+y+n+1)}\tag{5}$$
