# Falling and rising factorials

(Redirected from Pochhammer symbol)

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