Is there a reasonable generalization of the falling factorial for real exponents?

Question

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?

Answer 1

The gamma function naturally generalizes the factorial to complex values. It satisfies the functional equation $x\Gamma(x)=\Gamma(x+1)$ for any $x$ (when both sides exist anyway). Hence

$$\Gamma\big(x-(n-1)\big)\prod_{k=0}^{n-1}(x-k)=\Gamma(x+1)$$

by induction. Divide by the $\Gamma$ on the left and we're done.

Answer 2

$$ x^\underline n = \prod_{k=0}^{n-1}(x-k)= \frac{\Gamma(x+1)}{\Gamma(x+1-n)} $$