In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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,
[ (A reprint of the 1950 edition by Chelsea Publishing Co.)] rising sequential product, or upper factorial) is defined as
:
The value of each is taken to be 1 (an
empty product
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...
) when . These symbols are collectively called
factorial powers.
The Pochhammer symbol, introduced by
Leo August Pochhammer Leo August Pochhammer (25 August 1841, Stendal – 24 March 1920, Kiel) was a Prussian mathematician who was educated in Berlin, obtaining his Ph.D. in 1863 under Ernst Kummer. He became a lecturer in 1874, then professor of mathematics at the ...
, is the notation , where is a
non-negative integer
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal n ...
. It may represent ''either'' the rising or the falling factorial, with different articles and authors using different conventions. Pochhammer himself actually used with yet another meaning, namely to denote the
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
.
[. The remark about the Pochhammer symbol is on page 414.]
In this article, the symbol is used to represent the falling factorial, and the symbol is used for the rising factorial. These conventions are used in
combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
, although
Knuth's underline and overline notations
and
are increasingly popular.
In the theory of
special functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined by ...
(in particular the
hypergeometric function) and in the standard reference work ''
Abramowitz and Stegun
''Abramowitz and Stegun'' (''AS'') is the informal name of a 1964 mathematical reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards (NBS), now the ''National Institute of Standards and Te ...
'', the Pochhammer symbol is used to represent the rising factorial.
When is a positive integer, gives the number of
-permutations (sequences of distinct elements) from an -element set, or equivalently the number of
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
functions from a set of size to a set of size ; while gives the number of
partitions
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
of a -element set into ''x'' ordered sequences (possibly empty), or the number of ways to arrange ''k'' distinct flags on a row of ''x'' flagpoles.
Examples and combinatorial interpretation
The first few rising factorials are as follows:
:
The first few falling factorials are as follows:
:
The coefficients that appear in the expansions are
Stirling numbers of the first kind
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 poin ...
.
When the variable is a positive integer, the number is equal to the number of
-permutations from an -set, that is, the number of ways of choosing an ordered list of length consisting of distinct elements drawn from a collection of size . For example, is the number of different podiums—assignments of gold, silver, and bronze medals—possible in an eight-person race. Also, is "the number of ways to arrange flags on flagpoles",
where all flags must be used and each flagpole can have at most one flag. In this context, other notations like , or are also sometimes used.
Properties
The rising and falling factorials are simply related to one another:
:
The rising and falling factorials are directly related to the ordinary
factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial:
\begin
n! &= n \times (n-1) \times (n-2) \t ...
:
:
The rising and falling factorials can be used to express a
binomial coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the t ...
:
:
Thus many identities on binomial coefficients carry over to the falling and rising factorials.
The rising and falling factorials are well defined in any
unital ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
, and therefore can be taken to be, for example, a
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
, including negative integers, or a
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
with complex coefficients, or any
complex-valued function
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
.
The rising factorial can be extended to
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
values of using the
gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
provided and are real numbers that are not negative integers:
:
and so can the falling factorial:
:
Falling factorials appear in multiple
differentiation of simple power functions:
:
The rising factorial is also integral to the definition of the
hypergeometric function: The hypergeometric function is defined for by the
power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
:
provided that . Note, however, that the hypergeometric function literature typically uses the notation for rising factorials.
Relation to umbral calculus
The falling factorial occurs in a formula which represents
polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s using the forward
difference operator
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
, and which is formally similar to
Taylor's theorem
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
:
:
In this formula and in many other places, the falling factorial in the calculus of
finite difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...
s plays the role of in differential calculus. Note for instance the similarity of to .
A similar result holds for the rising factorial and the backward difference operator.
The study of analogies of this type is known as
umbral calculus
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain "shadowy" techniques used to "prove" them. These techniques were introduced by John Blis ...
. 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 sequence
In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are ...
s. Rising and falling factorials are Sheffer sequences of binomial type, as shown by the relations:
:
where the coefficients are the same as those in the
Binomial theorem
In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial into a sum involving terms of the form , where the ...
.
Similarly, the generating function of Pochhammer polynomials then amounts to the umbral exponential,
:
since
:
Connection coefficients and identities
The falling and rising factorials are related to one another through the
Lah numbers:
:
The following formulas relate integral powers of a variable through sums using the
Stirling numbers of the second kind
In mathematics, particularly in combinatorics, a Stirling number of the second kind (or Stirling partition number) is the number of ways to partition a set of ''n'' objects into ''k'' non-empty subsets and is denoted by S(n,k) or \textstyle \lef ...
, notated by curly brackets :
[
:
Since the falling factorials are a basis for the ]polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) ...
, one can express the product of two of them as a linear combination of falling factorials:
:
The coefficients are called ''connection coefficients'', and have a combinatorial interpretation as the number of ways to identify (or "glue together") elements each from a set of size and a set of size .
There is also a connection formula for the ratio of two rising factorials given by
:
Additionally, we can expand generalized exponent laws and negative rising and falling powers through the following identities:
:
Finally, duplication
Duplication, duplicate, and duplicator may refer to:
Biology and genetics
* Gene duplication, a process which can result in free mutation
* Chromosomal duplication, which can cause Bloom and Rett syndrome
* Polyploidy, a phenomenon also known ...
and multiplication formula
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addi ...
s for the falling and rising factorials provide the next relations:
:
Alternative notations
An alternative notation for the rising factorial
:
and for the falling factorial
:
goes back to A. Capelli (1893) and L. Toscano (1939), respectively. Graham, Knuth, and Patashnik propose to pronounce these expressions as " to the rising" and " to the falling", respectively.
Other notations for the falling factorial include , , , or . (See permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
and combination
In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are th ...
.)
An alternative notation for the rising factorial is the less common . When is used to denote the rising factorial, the notation is typically used for the ordinary falling factorial, to avoid confusion.[
]
Generalizations
The Pochhammer symbol has a generalized version called the generalized Pochhammer symbol In mathematics, the generalized Pochhammer symbol of parameter \alpha>0 and partition \kappa=(\kappa_1,\kappa_2,\ldots,\kappa_m) generalizes the classical Pochhammer symbol, named after Leo August Pochhammer, and is defined as
:(a)^_\kappa=\prod_ ...
, used in multivariate analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
. There is also a -analogue, the -Pochhammer symbol.
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:
:
where is the decrement and is the number of factors. The corresponding generalization of the rising factorial is
:
This notation unifies the rising and falling factorials, which are and respectively.
For any fixed arithmetic function and symbolic parameters , , related generalized factorial products of the form
:
may be studied from the point of view of the classes of generalized Stirling numbers of the first kind
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 poin ...
defined by the following coefficients of the powers of in the expansions of and then by the next corresponding triangular recurrence relation:
:
These coefficients satisfy a number of analogous properties to those for the Stirling numbers of the first kind
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 poin ...
as well as recurrence relations and functional equations related to the -harmonic numbers,
:
A symmetric generalization can be defined as
:
See also
* Pochhammer -symbol
*Vandermonde identity
In combinatorics, Vandermonde's identity (or Vandermonde's convolution) is the following identity for binomial coefficients:
:=\sum_^r
for any nonnegative integers ''r'', ''m'', ''n''. The identity is named after Alexandre-Théophile Vandermond ...
References
External links
*
Elementary Proofs
{{DEFAULTSORT:Pochhammer Symbol
Gamma and related functions
Factorial and binomial topics
Finite differences
Operations on numbers