Dixon's Theorem

${}_3F_2\left[{\matrix{n, -x, -y\cr x+n+1, y+n+1\cr}}\right]$
$=\Gamma(x+n+1)\Gamma(y+n+1)\Gamma({\textstyle{1\over 2}}n+1)\Gamma(x+y+{\textstyle{1\over 2}}n+1)$
$\times\Gamma(n+1)\Gamma(x+y+n+1)\Gamma(x+{\textstyle{1\over 2}}n+1)\Gamma(y+{\textstyle{1\over 2}}n+1),\quad$	(1)

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. It can be written more symmetrically as

$${}_3F_2(a,b,c;d,e;1)={({\textstyle{1\over 2}}a)!(a-b)!(a-c)!... ...extstyle{1\over 2}}a-b)!({\textstyle{1\over 2}}a-c)!(a-b-c)!},$$ (2)

where $1+a/2-b-c$ has a positive Real Part, $d=a-b+1$, and $e=a-c+1$. The identity can also be written as the beautiful symmetric sum

$$\sum_k (-1)^k{a+b\choose a+k}{a+c\choose c+k}{b+c\choose b+k}={(a+b+c)!\over a!b!c!}$$ (3)

(Petkovsek 1996).

See also Dougall-Ramanujan Identity, Generalized Hypergeometric Function

References

Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, 1935.

Cartier, P. and Foata, D. Problèmes combinatoires de commutation et réarrangements. New York: Springer-Verlag, 1969.

Knuth, D. E. The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 2nd ed. Reading, MA: Addison-Wesley, 1973.

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

Zeilberger, D. and Bressoud, D. ``A Proof of Andrew's $q$-Dyson Conjecture.'' Disc. Math. 54, 201-224, 1985.