# Definition:Rising Factorial

## Definition

Let $x$ be a real number (but usually an integer).

Let $n$ be a positive integer.

Then **$x$ to the (power of) $n$ rising** is defined as:

- $\ds x^{\overline n} := \prod_{j \mathop = 0}^{n - 1} \paren {x + j} = x \paren {x + 1} \cdots \paren {x + n - 1}$

## Also known as

This is referred to as the **$n$th rising factorial power of $x$**.

It can also be referred to as the **$n$th rising factorial of $x$**.

## Notation

The notation $x^{\overline n}$ for $x$ to the $n$ rising is due to Alfredo Capelli, who used it in $1893$.

This is the notation of choice on $\mathsf{Pr} \infty \mathsf{fWiki}$.

A more commonly seen notation (though arguably not as good) is $x^{\paren n}$.

This is known as the **Pochhammer function** or (together with $\paren x_n$ for its falling counterpart) the **Pochhammer symbol** (after Leo August Pochhammer).

However, depending on the context, either $\paren x_n$ or $x^{\paren n}$ can be used to indicate the rising factorial.

In the field of combinatorics $x^{\paren n}$ tends to be used, while in that of special functions you tend to see $\paren x_n$.

Therefore the more intuitively obvious $x^{\overline n}$ is becoming the preferred symbol for this.

See the note on notation in the Falling Factorial entry.

## Also see

- Results about
**rising factorials**can be found here.

## Sources

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.5$: Permutations and Factorials: $(19)$ - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**rising factorial**