Factorial

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

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

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

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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; Combination; Binomial coefficients.

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