---
tag: 79M1TG6F5N
kind: definition
name: Falling and rising factorials
msc: [05A10, 33B15]
defines: Y1BVY7J6AE
uses: [0N5M924N3P, 0WFCGVMRRC]
sources:
  - url: https://en.wikipedia.org/wiki/Falling_and_rising_factorials
---

## Definition

Let $x$ be an element of a commutative ring with identity, for example a real number, and let $n$ be a natural number.
The *falling factorial* of $x$ of order $n$ is
$$x^{\underline{n}} = x (x - 1) (x - 2) \cdots (x - n + 1) = \prod_{i=0}^{n-1} (x - i).$$
The *rising factorial* of $x$ of order $n$ is
$$x^{\overline{n}} = x (x + 1) (x + 2) \cdots (x + n - 1) = \prod_{i=0}^{n-1} (x + i).$$
For $n = 0$ both products are empty, and $x^{\underline{0}} = x^{\overline{0}} = 1$.

## Remarks

Example: for $x = 4$ and $n = 2$, $4^{\underline{2}} = 4 \cdot 3 = 12$ and $4^{\overline{2}} = 4 \cdot 5 = 20$.
Non-example: the product $4 \cdot 2 = 8$ is not $4^{\underline{2}}$, because its factors do not decrease by steps of $1$.
For a natural number $m \ge n$, $m^{\underline{n}} = m!/(m-n)!$, and $m^{\underline{n}} = 0$ when $n > m$; see [[falling factorial|0N5M924N3P]] and [[factorial|0WFCGVMRRC]].
For a natural number $m \ge 1$, $m^{\overline{n}} = (m + n - 1)!/(m - 1)!$, and $1^{\overline{n}} = n^{\underline{n}} = n!$.
The two are linked by $x^{\overline{n}} = (x + n - 1)^{\underline{n}} = (-1)^n (-x)^{\underline{n}}$.
For real $x > 0$, $x^{\overline{n}} = \Gamma(x + n)/\Gamma(x)$, where $\Gamma$ is the [[gamma function|0H5SYBSED9]].
Other notations: $(x)_n$ is used for the falling factorial in combinatorics and for the rising factorial (the Pochhammer symbol) in special functions; the rising factorial is also written $x^{(n)}$.
The [[q-Pochhammer symbol|Y1X8ZZQ7VM]] is a $q$-analogue of the rising factorial.