Generalizing the binomial coefficients to non integers

Question

We all know that ${n \choose k} = \dfrac{n!}{k!(n-k)!} $. But, when we are trying to evaluate expressions that involve the binomial expansion, we are sometimes lead to things like ${-1/3 \choose k }$. How do we make sense of it? In general, the question would be, what does

$$ {\alpha \choose k} $$

represent when $\alpha$ is in $\mathbb{R}$? How do we make sense of it and how do we compute it?

Answer 1

$$\binom{\alpha}k = \frac{\alpha(\alpha-1)(\alpha-2)\cdots (\alpha-k+1)}{k!}$$

You can see this in the Wikipedia article on binomial series, or in the binomial coefficient article under generalization and connection to the binomial series.

Answer 2

Perhaps as gamma functions? Using the fact that $\Gamma(x + 1) = x!$, we can write $$\dfrac{n!}{k!(n-k)!} = \dfrac{\Gamma(n+1)}{\Gamma(k+1)\Gamma(n-k+1)}$$ defined for $n > -1$, $k > -1$, $n > k -1$.