Falling and rising factorials

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In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial Falling and rising factorials_sentence_0

The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, rising sequential product, or upper factorial) is defined as Falling and rising factorials_sentence_1

The value of each is taken to be 1 (an empty product) when n = 0. Falling and rising factorials_sentence_2

These symbols are collectively called factorial powers. Falling and rising factorials_sentence_3

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. Falling and rising factorials_sentence_4

Also, (x)n is "the number of ways to arrange n flags on x flagpoles", where all flags must be used and each flagpole can have at most one flag. Falling and rising factorials_sentence_5

In this context, other notations like xPn and P(x, n) are also sometimes used. Falling and rising factorials_sentence_6

Examples Falling and rising factorials_section_0

The first few rising factorials are as follows: Falling and rising factorials_sentence_7

The first few falling factorials are as follows: Falling and rising factorials_sentence_8

The coefficients that appear in the expansions are Stirling numbers of the first kind. Falling and rising factorials_sentence_9

Properties Falling and rising factorials_section_1

The rising and falling factorials are simply related to one another: Falling and rising factorials_sentence_10

The rising and falling factorials are directly related to the ordinary factorial: Falling and rising factorials_sentence_11

The rising and falling factorials can be used to express a binomial coefficient: Falling and rising factorials_sentence_12

Thus many identities on binomial coefficients carry over to the falling and rising factorials. Falling and rising factorials_sentence_13

The rising and falling factorials are well defined in any unital ring, and therefore x can be taken to be, for example, a complex number, including negative integers, or a polynomial with complex coefficients, or any complex-valued function. Falling and rising factorials_sentence_14

The rising factorial can be extended to real values of n using the gamma function provided x and x + n are real numbers that are not negative integers: Falling and rising factorials_sentence_15

and so can the falling factorial: Falling and rising factorials_sentence_16

If D denotes differentiation with respect to x, one has Falling and rising factorials_sentence_17

The Pochhammer symbol is also integral to the definition of the hypergeometric function: The hypergeometric function is defined for |z| < 1 by the power series Falling and rising factorials_sentence_18

Relation to umbral calculus Falling and rising factorials_section_2

The falling factorial occurs in a formula which represents polynomials using the forward difference operator Δ and which is formally similar to Taylor's theorem: Falling and rising factorials_sentence_19

A similar result holds for the rising factorial. Falling and rising factorials_sentence_20

The study of analogies of this type is known as umbral calculus. Falling and rising factorials_sentence_21

A general theory covering such relations, including the falling and rising factorial functions, is given by the theory of polynomial sequences of binomial type and Sheffer sequences. Falling and rising factorials_sentence_22

Rising and falling factorials are Sheffer sequences of binomial type, as shown by the relations: Falling and rising factorials_sentence_23

where the coefficients are the same as the ones in the expansion of a power of a binomial (Chu–Vandermonde identity). Falling and rising factorials_sentence_24

Similarly, the generating function of Pochhammer polynomials then amounts to the umbral exponential, Falling and rising factorials_sentence_25

since Falling and rising factorials_sentence_26

Connection coefficients and identities Falling and rising factorials_section_3

The falling and rising factorials are related to one another through the Lah numbers: Falling and rising factorials_sentence_27

The following formulas relate integral powers of a variable x through sums using the Stirling numbers of the second kind ( notated by curly brackets { k} ): Falling and rising factorials_sentence_28

Since the falling factorials are a basis for the polynomial ring, one can express the product of two of them as a linear combination of falling factorials: Falling and rising factorials_sentence_29

There is also a connection formula for the ratio of two rising factorials given by Falling and rising factorials_sentence_30

Additionally, we can expand generalized exponent laws and negative rising and falling powers through the following identities: Falling and rising factorials_sentence_31

Finally, duplication and multiplication formulas for the rising factorials provide the next relations: Falling and rising factorials_sentence_32

Alternate notations Falling and rising factorials_section_4

An alternate notation for the rising factorial Falling and rising factorials_sentence_33

and for the falling factorial Falling and rising factorials_sentence_34

goes back to A. Capelli (1893) and L. Toscano (1939), respectively. Falling and rising factorials_sentence_35

Graham, Knuth, and Patashnik propose to pronounce these expressions as "x to the m rising" and "x to the m falling", respectively. Falling and rising factorials_sentence_36

Other notations for the falling factorial include P(x, n) , Pn , Px,n , or xPn . Falling and rising factorials_sentence_37

(See permutation and combination.) Falling and rising factorials_sentence_38

An alternate notation for the rising factorial x is the less common (x) n . Falling and rising factorials_sentence_39

When (x) n is used to denote the rising factorial, the notation (x) n is typically used for the ordinary falling factorial, to avoid confusion. Falling and rising factorials_sentence_40

Generalizations Falling and rising factorials_section_5

The Pochhammer symbol has a generalized version called the generalized Pochhammer symbol, used in multivariate analysis. Falling and rising factorials_sentence_41

There is also a q-analogue, the q-Pochhammer symbol. Falling and rising factorials_sentence_42

A generalization of the falling factorial in which a function is evaluated on a descending arithmetic sequence of integers and the values are multiplied is: Falling and rising factorials_sentence_43

where −h is the decrement and k is the number of factors. Falling and rising factorials_sentence_44

The corresponding generalization of the rising factorial is Falling and rising factorials_sentence_45

This notation unifies the rising and falling factorials, which are [x] and [x], respectively. Falling and rising factorials_sentence_46

See also Falling and rising factorials_section_6

Falling and rising factorials_unordered_list_0

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Falling and rising factorials.