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Difference between revisions of "Factorial"

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(Comments: rising and falling)
(→‎Comments: Pochhammer symbol)
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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|Combinatorial analysis]]; [[Combination]]; [[Binomial coefficients]].
 
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|Combinatorial analysis]]; [[Combination]]; [[Binomial coefficients]].
  
The quantity $(a)_\mu$ is a ''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)\cdots(a-\mu+1) \ .
 
$$
 
$$

Revision as of 21:07, 16 January 2016

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. The factorial is equal to the number of permutations of $n$ elements. The more general expression

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

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 $$ (a)^{\underline{\mu}} = a(a-1)\cdots(a-\mu+1) \ . $$

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