Gamma and related functions | Operations on numbers | Finite differences | Factorial and binomial topics

Falling and rising factorials

In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, rising sequential product, or upper factorial) is defined as The value of each is taken to be 1 (an empty product) when n = 0. These symbols are collectively calledfactorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation (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 (x)n with yet another meaning, namely to denote the binomial coefficient . In this article, the symbol (x)n is used to represent the falling factorial, and the symbol x(n) is used for the rising factorial. These conventions are used in combinatorics, although Knuth's underline and overline notations and are increasingly popular. In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and Stegun, the Pochhammer symbol (x)n is used to represent the rising factorial. When x is a positive integer, (x)n gives the number of n-permutations of an x-element set, or equivalently the number of injective functions from a set of size n to a set of size x. (Wikipedia).

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From playlist Factoring Trinomials with a Leading Coefficient Not 1

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how to simplify an expression raised to a negative power

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Related pages

Abramowitz and Stegun | Q-analog | Derivative | Sheffer sequence | Mathematical analysis | Empty product | Q-Pochhammer symbol | Taylor's theorem | Finite difference | Permutation | Polynomial | Hypergeometric function | Umbral calculus | Special functions | Combinatorics | Factorial | Polynomial ring | Generalized Pochhammer symbol | Injective function | Binomial type | Stirling numbers of the second kind | Gamma function | Binomial coefficient | Mathematics | Real number | Power series | Ring (mathematics) | Stirling numbers of the first kind | Linear combination | Complex number | Combination | Pochhammer k-symbol