The falling factorial is defined as:
$$x^\underline n = \prod_{k=0}^{n-1}(x-k),\quad n\in\mathbb N$$
It can be used to define the binomial:
$$\binom n k = \frac{n^\underline k}{k!}$$
And it satisfies identities for discrete differential calculus as $x^a$ does for continuous differential calculus. $\Delta$ is the difference operator $\left(\Delta a_n = \frac{a_{n+1}-a_n}1\right)\colon$
$$\Delta x^\underline k = kx^\underline{k-1}.$$
One can easily find a reasonable generalization for arbitrary $n\in\mathbb Z$, but I wasn't able to come up with a generalization for $n\in\mathbb R$ or even $n\in\mathbb C$. Is there one?