Rising and Falling Factorials

# Rising and Falling Factorials

Recall from The Factorial Function page that if $n \in \{1, 2, ... \}$ then:

(1)
\begin{align} \quad n! = n \cdot (n - 1) \cdot ... \cdot 2 \cdot 1 \end{align}

We also conventionally defined $0! = 1$.

We will now look at two other types of factorials - naming the rising factorial and the falling factorial which we define below.

 Definition: For all nonnegative integers $x$, the Rising Factorial $x^{\overline{n}}$ is defined to be $x^{\overline{n}} = \underbrace{x \cdot (x + 1) \cdot ... \cdot (x + n - 1)}_{n \: \mathrm{factors}}$.

For example:

(2)
\begin{align} \quad 2^{\overline{3}} = 2 \cdot 3 \cdot 4 = 24 \end{align}
 Definition: For all nonnegative integers $x$, the Falling Factorial $x^{\underline{n}}$ is defined to be $x^{\underline{n}} = \underbrace{x \cdot (x - 1) \cdot ... \cdot (x - n + 1)}_{n \: \mathrm{factors}}$.

For example:

(3)
\begin{align} \quad 5^{\underline{3}} = 5 \cdot 4 \cdot 3 = 60 \end{align}