Editing Falling and rising factorials (section)

==Properties==

The rising and falling factorials are simply related to one another:
<math display="block">
\begin{alignat}{2}
{(x)}_n &= {(x-n+1)}^{(n)} &&= (-1)^n (-x)^{(n)},\\
x^{(n)} &= {(x+n-1)}_{n} &&= (-1)^n (-x)_n.
\end{alignat}</math>

Falling and rising factorials of integers are directly related to the ordinary [[factorial]]:
<math display="block">
\begin{align}
n! &= 1^{(n)} = (n)_n,\\[6pt]
(m)_n &= \frac{m!}{(m-n)!},\\[6pt]
m^{(n)} &= \frac{(m+n-1)!}{(m-1)!}.
\end{align}</math>

Rising factorials of half integers are directly related to the [[double factorial]]:
<math display="block">
\begin{align}
\left[\frac{1}{2}\right]^{(n)} = \frac{(2n-1)!!}{2^n},\quad
\left[\frac{2m+1}{2}\right]^{(n)} = \frac{(2(n+m)-1)!!}{2^n(2m-1)!!}.
\end{align}</math>

The falling and rising factorials can be used to express a [[binomial coefficient]]:
<math display="block">
\begin{align}
\frac{(x)_n}{n!} &= \binom{x}{n},\\[6pt]
\frac{x^{(n)}}{n!} &= \binom{x+n-1}{n}.
\end{align}</math>

Thus many identities on binomial coefficients carry over to the falling and rising factorials.

The rising and falling factorials are well defined in any [[Unital ring|unital]] [[ring (mathematics)|ring]], and therefore <math>x</math> can be taken to be, for example, a [[complex number]], including negative integers, or a [[polynomial]] with complex coefficients, or any [[complex-valued function]].

===Real numbers and negative ''n''===
The falling factorial can be extended to [[real number|real]] values of <math>x</math> using the [[gamma function]] provided <math>x</math> and <math>x+n</math> are real numbers that are not negative integers:
<math display="block">
(x)_n = \frac{\Gamma(x+1)}{\Gamma(x-n+1)}\ ,
</math>
and so can the rising factorial:
<math display="block">
x^{(n)} = \frac{\Gamma(x+n)}{\Gamma(x)}\ .
</math>

===Calculus===

Falling factorials appear in multiple [[derivative|differentiation]] of simple power functions:
<math display="block">
\left(\frac{\mathrm{d}}{\mathrm{d}x}\right)^n x^a = (a)_n \cdot x^{a-n}.
</math>

The rising factorial is also integral to the definition of the [[hypergeometric function]]: The hypergeometric function is defined for <math>|z| < 1</math> by the [[power series]]
<math display="block">
{}_2F_1(a,b;c;z) = \sum_{n=0}^\infty \frac{a^{(n)} b^{(n)}}{c^{(n)}}  \frac{z^n}{n!}
</math>
provided that <math>c \neq 0, -1, -2, \ldots</math>. Note, however, that the hypergeometric function literature typically uses the notation <math>(a)_n</math> for rising factorials.