Falling and rising factorials

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

{\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

{\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 ${\tbinom {x}{n}}.$

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 $x^{\underline {n}}$ and $x^{\overline {n}}$ 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 $x$ . The rising factorial $x (n)$ gives the number of partitions of an $n$ -element set into $x$ ordered sequences (possibly empty).

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