Let $X$ be a random variable following a beta-binomial distribution:
$$ X \sim \mathrm{BetBin}(n, \alpha, \beta) \; . $$
According to Wikipedia, the cumulative distribution function of $X$ is
$$ F_X(x) = {n \choose x} \frac{\mathrm{B}(x+\alpha,n-x+\beta)}{\mathrm{B}(\alpha,\beta)} {}_3 F_2(a; b; x) $$
where ${}_3 F_2(a, b; x)$ is supposed to be the generalized hypergeometric function
$$ {}_3 F_2([1, -x, n-x+\beta]; [n-x+1, 1-x-\alpha]; 1) \; . $$
Trying to derive the CDF by hand, i.e. as
$$ F_X(x) = \mathrm{Pr}(X \leq x) = \sum_{i=-\infty}^{x} f_X(i) = \sum_{i=0}^{x} {n \choose i} \cdot \frac{\mathrm{B}(\alpha+i,\beta+n-i)}{\mathrm{B}(\alpha,\beta)} \; , $$
I have managed to get up to this point:
$$ F_X(x) = \frac{1}{\mathrm{B}(\alpha,\beta)} \cdot \frac{\Gamma(n+1)}{\Gamma(\alpha+\beta+n)} \cdot \sum_{i=0}^{x} \frac{\Gamma(\alpha+i) \cdot \Gamma(\beta+n-i)}{\Gamma(i+1) \cdot \Gamma(n-i+1)} \; . $$
Can somebody tell me how to arrive at the equation that is given on Wikipedia?