Dixon's identity

Statement

1. For every nonnegative integer $a$, the binomial coefficients satisfy $$\sum_{k = -a}^{a} (-1)^k \binom{2a}{a + k}^3 = \frac{(3a)!}{(a!)^3}.$$
2. For nonnegative integers $a, b, c$, with $k$ running over all integers and $\binom{n}{j}$ read as $0$ for $j < 0$ or $j > n$, $$\sum_{k} (-1)^k \binom{a + b}{a + k} \binom{b + c}{b + k} \binom{c + a}{c + k} = \frac{(a + b + c)!}{a!\, b!\, c!}.$$
3. Let $a, b, c$ be complex numbers such that $1 + a - b$, $1 + a - c$ and $1 + \frac{a}{2}$ are not in $\set{0, -1, -2, \dots}$ and $\operatorname{Re}\left( \frac{a}{2} - b - c \right) > -1$. Then, with $\Gamma$ the gamma function and ${}_3F_2$ the generalized hypergeometric series, $${}_3F_2\left( \begin{matrix} a,\ b,\ c \\ 1 + a - b,\ 1 + a - c \end{matrix} ; 1 \right) = \frac{\Gamma(1 + \frac{a}{2})\, \Gamma(1 + a - b)\, \Gamma(1 + a - c)\, \Gamma(1 + \frac{a}{2} - b - c)}{\Gamma(1 + a)\, \Gamma(1 + \frac{a}{2} - b)\, \Gamma(1 + \frac{a}{2} - c)\, \Gamma(1 + a - b - c)}.$$