Gauss's Hypergeometric Theorem

$${}_2F_1(a,b;c;1) = {\Gamma(c)\Gamma(c-a-b)\over \Gamma(c-a)\Gamma(c-b)}$$

for $\Re[c-a-b]>0$, where ${}_2F_1(a,b;c;x)$ is a Hypergeometric Function. If $a$ is a Negative Integer $-n$, this becomes

$${}_2F_1(-n,b;c;1) = {(c-b)_n\over (c)_n},$$

which is known as the Vandermonde Theorem.

See also Generalized Hypergeometric Function, Hypergeometric Function

References

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, pp. 42 and 126, 1996.