Thomae's Theorem

${\Gamma(x+y+s+1)\over\Gamma(x+s+1)\Gamma(y+s+1)} {}_3F_2\left({\matrix{-a, -b, x+y+s+1\cr x+s+1, y+s+1}; 1}\right)$
$={\Gamma(a+b+s+1)\over \Gamma(a+s+1)\Gamma(b+s+1)} {}_3F_2\left({\matrix{-x, -y, a+b+s+1\cr a+s+1, b+s+1}; 1}\right),$

where $\Gamma(z)$ is the Gamma Function and the function ${}_3F_2(a,b,c;d,e;z)$ is a Generalized Hypergeometric Function.

See also Generalized Hypergeometric Function

References

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 104-105, 1959.