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The function defined on the set of non-negative integers with value at $n$ equal to the product of the natural numbers from 1 to $n$, that is, to $1\cdot2\ldots n$; it is denoted by $n!$ (by definition, $0!=1$). For large $n$ an approximate expression for the factorial is given by the [[Stirling formula|Stirling formula]]. The factorial is equal to the number of permutations of $n$ elements. The more general expression
+
The function defined on the set of non-negative integers with value at $n$ equal to the product of the natural numbers from 1 to $n$, that is, to $1\cdot2\cdot\ldots\cdot n$; it is denoted by $n!$ (by definition, $0!=1$). For large $n$ an approximate expression for the factorial is given by the [[Stirling formula|Stirling formula]]. The factorial is equal to the number of permutations of $n$ elements. The more general expression
  
$$(a)_\mu=a(a+1)\cdots(a+\mu-1) \ ,$$
+
$$(a)_\mu=a(a+1)\dots(a+\mu-1) \ ,$$
  
 
is also called a factorial, where $a$ is a complex number, $\mu$ is a natural number, and $(a)_0=1$. See also [[Gamma-function|Gamma-function]].
 
is also called a factorial, where $a$ is a complex number, $\mu$ is a natural number, and $(a)_0=1$. See also [[Gamma-function|Gamma-function]].
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The ''Pochhammer symbol'' $(a)_\mu$ denotes the ''rising factorial'', also denoted $(a)^{\overline{\mu}}$.  Analogously, one defines the ''falling factorial''
 
The ''Pochhammer symbol'' $(a)_\mu$ denotes the ''rising factorial'', also denoted $(a)^{\overline{\mu}}$.  Analogously, one defines the ''falling factorial''
 
$$
 
$$
(a)^{\underline{\mu}} = a(a-1)\cdots(a-\mu+1) \ .
+
(a)^{\underline{\mu}} = a(a-1)\dots(a-\mu+1) \ .
 
$$
 
$$

Revision as of 14:15, 30 December 2018

The function defined on the set of non-negative integers with value at $n$ equal to the product of the natural numbers from 1 to $n$, that is, to $1\cdot2\cdot\ldots\cdot n$; it is denoted by $n!$ (by definition, $0!=1$). For large $n$ an approximate expression for the factorial is given by the Stirling formula. The factorial is equal to the number of permutations of $n$ elements. The more general expression

$$(a)_\mu=a(a+1)\dots(a+\mu-1) \ ,$$

is also called a factorial, where $a$ is a complex number, $\mu$ is a natural number, and $(a)_0=1$. See also Gamma-function.


Comments

Because $n!$ equals the number of permutations of $n$ elements, the factorial is extensively used in combinatorics, probability theory, mathematical statistics, etc. Cf. Combinatorial analysis; Combination; Binomial coefficients.

The Pochhammer symbol $(a)_\mu$ denotes the rising factorial, also denoted $(a)^{\overline{\mu}}$. Analogously, one defines the falling factorial $$ (a)^{\underline{\mu}} = a(a-1)\dots(a-\mu+1) \ . $$

How to Cite This Entry:
Factorial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Factorial&oldid=37568
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article