Dougall's Theorem

${}_5F_4\left[{\matrix{{\textstyle{1\over 2}}n+1, n, -x, -y, -z\cr {\textstyle{1\over 2}}n, x+n+1, y+n+1, z+n+1\cr}}\right]$

$={\Gamma(x+n+1)\Gamma(y+n+1)\Gamma(z+n+1)\Gamma(x+y+z+n+1)\over \Gamma(n+1)\Gamma(x+y+n+1)\Gamma(y+z+n+1)\Gamma(x+z+n+1)},$

where ${}_5F_4(a,b,c,d,e;f,g,h,i;z)$ is a Generalized Hypergeometric Function and $\Gamma(z)$ is the Gamma Function.

See also Dougall-Ramanujan Identity, Generalized Hypergeometric Function

${}_5F_4\left[{\matrix{{\textstyle{1\over 2}}n+1, n, -x, -y, -z\cr {\textstyle{1\over 2}}n, x+n+1, y+n+1, z+n+1\cr}}\right]$
$={\Gamma(x+n+1)\Gamma(y+n+1)\Gamma(z+n+1)\Gamma(x+y+z+n+1)\over \Gamma(n+1)\Gamma(x+y+n+1)\Gamma(y+z+n+1)\Gamma(x+z+n+1)},$