Difference between revisions of "Factorial"
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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\ | + | 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 $1\cdot2\dots n$; 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 |
| − | $$(a)_\mu=a(a+1)\ | + | $$(a)_\mu=a(a+1)\dots(a+\mu-1) \ ,$$ |
is also called a factorial, where $a$ is a complex number, $\mu$ is a natural number, and $(a)_0=1$. See also [[Gamma-function|Gamma-function]]. | is also called a factorial, where $a$ is a complex number, $\mu$ is a natural number, and $(a)_0=1$. See also [[Gamma-function|Gamma-function]]. | ||
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The ''Pochhammer symbol'' $(a)_\mu$ denotes the ''rising factorial'', also denoted $(a)^{\overline{\mu}}$. Analogously, one defines the ''falling factorial'' | The ''Pochhammer symbol'' $(a)_\mu$ denotes the ''rising factorial'', also denoted $(a)^{\overline{\mu}}$. Analogously, one defines the ''falling factorial'' | ||
$$ | $$ | ||
| − | (a)^{\underline{\mu}} = a(a-1)\ | + | (a)^{\underline{\mu}} = a(a-1)\dots(a-\mu+1) \ . |
$$ | $$ | ||
Revision as of 13:57, 30 December 2018
The function defined on the set of non-negative integers with value at $n$ equal to the product of the natural numbers from 1 to $n$, that is, to $1\cdot2\dots n$; it is denoted by $n!$ (by definition, $0!=1$). For large $n$ an approximate expression for the factorial is given by the Stirling formula. The factorial is equal to the number of permutations of $n$ elements. The more general expression
$$(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) \ . $$
Factorial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Factorial&oldid=37568