Double factorial

Double factorial
In mathematics, the double factorial of a number , denoted by , is the product of all the positive integers up to that have the same parity (odd or even) as . That is,
Restated, this says that for even , the double factorial is
while for odd it is
For example, . The zero double factorial as an empty product.
The sequence of double factorials for even = starts as
The sequence of double factorials for odd = starts as
The term odd factorial is sometimes used for the double factorial of an odd number.
History and usage
In a 1902 paper, the physicist Arthur Schuster wrote:
states that the double factorial was originally introduced in order to simplify the expression of certain trigonometric integrals that arise in the derivation of the Wallis product. Double factorials also arise in expressing the volume of a hypersphere, and they have many applications in enumerative combinatorics. They occur in Student's -distribution (1908), though Gosset did not use the double exclamation point notation.
Relation to the factorial
Because the double factorial only involves about half the factors of the ordinary factorial, its value is not substantially larger than the square root of the factorial , and it is much smaller than the iterated factorial .
The factorial of a positive may be written as the product of two double factorials:
and therefore
where the denominator cancels the unwanted factors in the numerator. (The last form also applies when .)
For an even non-negative integer with , the double factorial may be expressed as
For odd with , combining the two previous formulas yields
For an odd positive integer with , the double factorial may be expressed in terms of -permutations of or a falling factorial as
Applications in enumerative combinatorics
Double factorials are motivated by the fact that they occur frequently in enumerative combinatorics and other settings. For instance, for odd values of counts
Perfect matchings of the complete graph for odd . In such a graph, any single vertex v has possible choices of vertex that it can be matched to, and once this choice is made the remaining problem is one of selecting a perfect matching in a complete graph with two fewer vertices. For instance, a complete graph with four vertices a, b, c, and d has three perfect matchings: ab and cd, ac and bd, and ad and bc. Perfect matchings may be described in several other equivalent ways, including involutions without fixed points on a set of items (permutations in which each cycle is a pair) or chord diagrams (sets of chords of a set of points evenly spaced on a circle such that each point is the endpoint of exactly one chord, also called Brauer diagrams). The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are instead given by the telephone numbers, which may be expressed as a summation involving double factorials.
Stirling permutations, permutations of the multiset of numbers in which each pair of equal numbers is separated only by larger numbers, where . The two copies of must be adjacent; removing them from the permutation leaves a permutation in which the maximum element is , with positions into which the adjacent pair of values may be placed. From this recursive construction, a proof that the Stirling permutations are counted by the double permutations follows by induction. Alternatively, instead of the restriction that values between a pair may be larger than it, one may also consider the permutations of this multiset in which the first copies of each pair appear in sorted order; such a permutation defines a matching on the positions of the permutation, so again the number of permutations may be counted by the double permutations.
Heap-ordered trees, trees with nodes labeled , such that the root of the tree has label 0, each other node has a larger label than its parent, and such that the children of each node have a fixed ordering. An Euler tour of the tree (with doubled edges) gives a Stirling permutation, and every Stirling permutation represents a tree in this way.
Unrooted binary trees with labeled leaves. Each such tree may be formed from a tree with one fewer leaf, by subdividing one of the tree edges and making the new vertex be the parent of a new leaf.
Rooted binary trees with labeled leaves. This case is similar to the unrooted case, but the number of edges that can be subdivided is even, and in addition to subdividing an edge it is possible to add a node to a tree with one fewer leaf by adding a new root whose two children are the smaller tree and the new leaf.
and list several additional objects with the same counting sequence, including "trapezoidal words" (numerals in a mixed radix system with increasing odd radixes), height-labeled Dyck paths, height-labeled ordered trees, "overhang paths", and certain vectors describing the lowest-numbered leaf descendant of each node in a rooted binary tree. For bijective proofs that some of these objects are equinumerous, see and .
The even double factorials give the numbers of elements of the hyperoctahedral groups (signed permutations or symmetries of a hypercube)
Asymptotics
Stirling's approximation for the factorial can be used to derive an asymptotic equivalent for the double factorial. In particular, since one has as tends to infinity that
Extensions
Negative arguments
The ordinary factorial, when extended to the gamma function, has a pole at each negative integer, preventing the factorial from being defined at these numbers. However, the double factorial of odd numbers may be extended to any negative odd integer argument by inverting its recurrence relation
to give
Using this inverted recurrence, (−1)!! = 1, (−3)!! = −1, and (−5)!! = ; negative odd numbers with greater magnitude have fractional double factorials. In particular, when is an odd number, this gives
Complex arguments
Disregarding the above definition of for even values of , the double factorial for odd integers can be extended to most real and complex numbers by noting that when is a positive odd integer then
where is the gamma function.
The final expression is defined for all complex numbers except the negative even integers and satisfies everywhere it is defined. As with the gamma function that extends the ordinary factorial function, this double factorial function is logarithmically convex in the sense of the Bohr–Mollerup theorem. Asymptotically,
The generalized formula does not agree with the previous product formula for for non-negative even integer values of . Instead, this generalized formula implies the following alternative:
with the value for 0!! in this case being
.
Using this generalized formula as the definition, the volume of an -dimensional hypersphere of radius can be expressed as
Additional identities
For integer values of ,
Using instead the extension of the double factorial of odd numbers to complex numbers, the formula is
Double factorials can also be used to evaluate integrals of more complicated trigonometric polynomials.
Double factorials of odd numbers are related to the gamma function by the identity:
Some additional identities involving double factorials of odd numbers are:
An approximation for the ratio of the double factorial of two consecutive integers is
This approximation gets more accurate as increases, which can be seen as a result of the Wallis Integral.
Generalizations
Definitions
In the same way that the double factorial generalizes the notion of the single factorial, the following definition of the integer-valued multiple factorial functions (multifactorials), or -factorial functions, extends the notion of the double factorial function for positive integers :
Alternative extension of the multifactorial
Alternatively, the multifactorial can be extended to most real and complex numbers by noting that when is one more than a positive multiple of the positive integer then
This last expression is defined much more broadly than the original. In the same way that is not defined for negative integers, and is not defined for negative even integers, is not defined for negative multiples of . However, it is defined for and satisfies for all other complex numbers . This definition is consistent with the earlier definition only for those integers satisfying .
In addition to extending to most complex numbers , this definition has the feature of working for all positive real values of . Furthermore, when , this definition is mathematically equivalent to the function, described above. Also, when , this definition is mathematically equivalent to the alternative extension of the double factorial.
Generalized Stirling numbers expanding the multifactorial functions
A class of generalized Stirling numbers of the first kind is defined for by the following triangular recurrence relation:
These generalized -factorial coefficients then generate the distinct symbolic polynomial products defining the multiple factorial, or -factorial functions as
The distinct polynomial expansions in the previous equations actually define the -factorial products for multiple distinct cases of the least residues for .
The generalized -factorial polynomials, where , which generalize the Stirling convolution polynomials from the single factorial case to the multifactorial cases, are defined by
for . These polynomials have a particularly nice closed-form ordinary generating function given by
Other combinatorial properties and expansions of these generalized -factorial triangles and polynomial sequences are considered in .
Exact finite sums involving multiple factorial functions
Suppose that and are integer-valued. Then we can expand the next single finite sums involving the multifactorial, or -factorial functions in terms of the Pochhammer symbol and the generalized, rational-valued binomial coefficients as
and moreover, we similarly have double sum expansions of these functions given by
The first two sums above are similar in form to a known non-round combinatorial identity for the double factorial function when given by .
Similar identities can be obtained via context-free grammars. Additional finite sum expansions of congruences for the -factorial functions modulo any prescribed integer for any are given by .
References
Integer sequences
Enumerative combinatorics
Factorial and binomial topics
fr:Analogues de la factorielle#Multifactorielles