In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial

${\displaystyle {\begin{aligned}(x)_{n}=x^{\underline {n}}&=\overbrace {x(x-1)(x-2)\cdots (x-n+1)} ^{n{\text{ factors}}}\\&=\prod _{k=1}^{n}(x-k+1)=\prod _{k=0}^{n-1}(x-k)\,.\end{aligned}}}$

The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, rising sequential product, or upper factorial) is defined as

${\displaystyle {\begin{aligned}x^{(n)}=x^{\overline {n}}&=\overbrace {x(x+1)(x+2)\cdots (x+n-1)} ^{n{\text{ factors}}}\\&=\prod _{k=1}^{n}(x+k-1)=\prod _{k=0}^{n-1}(x+k)\,.\end{aligned}}}$

The value of each is taken to be 1 (an empty product) when n = 0 . These symbols are collectively called factorial powers.

The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation (x)_n , where n is a non-negative integer. It may represent either the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used (x)_n with yet another meaning, namely to denote the binomial coefficient ${\displaystyle \ {\tbinom {x}{n}}\ .}$

In this article, the symbol (x)_n is used to represent the falling factorial, and the symbol x⁽ⁿ⁾ is used for the rising factorial. These conventions are used in combinatorics, although Knuth's underline and overline notations ${\displaystyle \ x^{\underline {n}}\ }$ and ${\displaystyle \ x^{\overline {n}}\ }$ are increasingly popular. In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and Stegun, the Pochhammer symbol (x)_n is used to represent the rising factorial.

When x is a positive integer, (x)_n gives the number of n-permutations (sequences of distinct elements) from an x-element set, or equivalently the number of injective functions from a set of size n to a set of size x; while x⁽ⁿ⁾ gives the number of partitions of a k-element set into x ordered sequences (possibly empty), or the number of ways to arrange k distinct flags on a row of x flagpoles.