Is there any simple proof of this identity
$${_4F_3}[\{\frac{1}{2}+\frac{n}{4},1+\frac{n}{4},-p,p\},\{\frac{1}{2},\frac{3}{2},\frac{1}{2}+\frac{n}{2}\},1]=\frac{\Gamma[\frac{1+n}{2}] (\frac{\cos[\frac{n \pi }{2}] \Gamma[\frac{1}{2}-\frac{n}{2}+p]}{\Gamma[\frac{3}{2}+p]}-\frac{\Gamma[-\frac{1}{2}+p]}{\Gamma[\frac{1+n}{2}+p]})}{4 \sqrt{\pi }}$$
Here $p$ and $n$ are certain positive integers. It seems that the formula is true only for some integers...