Falling and rising factorials

In mathematics, the falling factorial (sometimes called the descending factorial,^[1] 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,^[1] 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 ${\displaystyle n=0}$. These symbols are collectively called factorial powers.^[2]

The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation ${\displaystyle (x)_{n}}$, 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 ${\displaystyle (x)_{n}}$ with yet another meaning, namely to denote the binomial coefficient ${\displaystyle {\tbinom {x}{n}}}$.^[3]

In this article, the symbol ${\displaystyle (x)_{n}}$ is used to represent the falling factorial, and the symbol ${\displaystyle x^{(n)}}$ is used for the rising factorial. These conventions are used in combinatorics,^[4] although Knuth's underline and overline notations ${\displaystyle x^{\underline {n}}}$ and ${\displaystyle x^{\overline {n}}}$ are increasingly popular.^[2][5] In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and Stegun, the Pochhammer symbol ${\displaystyle (x)_{n}}$ is used to represent the rising factorial.^[6][7]

When ${\displaystyle x}$ is a positive integer, ${\displaystyle (x)_{n}}$ 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 ${\displaystyle n}$ to a set of size ${\displaystyle x}$. The rising factorial ${\displaystyle x^{(n)}}$ gives the number of partitions of an ${\displaystyle n}$-element set into ${\displaystyle x}$ ordered sequences (possibly empty).^[a]

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