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 These symbols are collectively called factorial 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 (sequences of distinct elements) from an x-element set, or equivalently the number of injective functions from a set of size n to a set of size x. The rising factorial x^(n) gives the number of partitions of an n-element set into x ordered sequences (possibly empty).
The first few falling factorials are as follows:
The first few rising factorials are as follows:
The coefficients that appear in the expansions are Stirling numbers of the first kind (see below).
When the variable x is a positive integer, the number (x)_n is equal to the number of n-permutations from a set of x items, that is, the number of ways of choosing an ordered list of length n consisting of distinct elements drawn from a collection of size x.
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vignette La formule de Stirling, du nom du mathématicien écossais James Stirling, donne un équivalent de la factorielle d'un entier naturel n quand n tend vers l'infini : que l'on trouve souvent écrite ainsi : où le nombre e désigne la base de l'exponentielle. C'est Abraham de Moivre qui a initialement démontré la formule suivante : où C est une constante réelle (non nulle). L'apport de Stirling fut d'attribuer la valeur C = à la constante et de donner un développement de ln(n!) à tout ordre.
En mathématiques, et notamment en analyse et en combinatoire, une série génératrice (appelée autrefois fonction génératrice, terminologie encore utilisée en particulier dans le contexte de la théorie des probabilités) est une série formelle dont les coefficients codent une suite de nombres (ou plus généralement de polynômes) ; on dit que la série est associée à la suite. Ces séries furent introduites par Abraham de Moivre en 1730, pour obtenir des formules explicites pour des suites définies par récurrence linéaire.
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one). The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of the first kind.
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