Saalschütz's Theorem

$${}_3F_2\left[{\matrix{-x, -y, -z\cr n+1, -x-y-z\cr}}\right] ... ...+n+1)\Gamma(z+x+n+1)\over\Gamma(z+n+1)\Gamma(x+y+z+n+1)},\eqnm$$

where ${}_3F_2(a,b,c;d,e;z)$ is a Generalized Hypergeometric Function and $\Gamma(z)$ is the Gamma Function. It can be derived from the Dougall-Ramanujan Identity and written in the symmetric form

$${}_3F_2(a,b,c;d,e;1)={(d-a)_{\vert c\vert}(d-b)_{\vert c\vert}\over d_{\vert c\vert}(d-a-b)_{\vert c\vert}}$$

for $d+e=a+b+c+1$ with $c$ a negative integer and $(a)_n$ the Pochhammer Symbol (Petkovsek et al. 1996).

See also Dougall-Ramanujan Identity, Generalized Hypergeometric Function

References

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, pp. 43 and 126, 1996.