The original identity, from , is :\sum_{k=-a}^{a}(-1)^{k}{2a\choose k+a}^3 =\frac{(3a)!}{(a!)^3}. A generalization, also sometimes called Dixon's identity, is :\sum_{k\in\mathbb{Z}}(-1)^k{a+b\choose a+k} {b+c\choose b+k}{c+a\choose c+k} = \frac{(a+b+c)!}{a!b!c!} where
a,
b, and
c are non-negative
integers . The sum on the left can be written as the terminating well-poised hypergeometric series :{b+c\choose b-a}{c+a\choose c-a}{}_3F_2(-2a,-a-b,-a-c;1+b-a,1+c-a;1) and the identity follows as a limiting case (as
a tends to an integer) of Dixon's theorem evaluating a well-poised 3
F2
generalized hypergeometric series at 1, from : :\;_3F_2 (a,b,c;1+a-b,1+a-c;1)= \frac{\Gamma(1+a/2)\Gamma(1+a/2-b-c)\Gamma(1+a-b)\Gamma(1+a-c)} {\Gamma(1+a)\Gamma(1+a-b-c)\Gamma(1+a/2-b)\Gamma(1+a/2-c)}. This holds for Re(1 +
a −
b −
c) > 0. As
c tends to −∞ it reduces to
Kummer's formula for the hypergeometric function 2F1 at −1. Dixon's theorem can be deduced from the evaluation of the
Selberg integral. ==
q-analogues==