View source for Double factorial

[[File:Chord diagrams K6 matchings.svg|thumb|360px|The fifteen different chord diagrams on six points, or equivalently the fifteen different [[perfect matching]]s on a six-vertex [[complete graph]].  These are counted by the double factorial {{math|15 {{=}} (6 − 1)!!}}.]]
In [[mathematics]], the '''double factorial''' or '''semifactorial''' of a number {{mvar|n}} (denoted by {{math|''n''!!}}) is the product of all the integers from 1 up to {{mvar|n}} that have the same [[parity (mathematics)|parity (odd or even)]] as {{mvar|n}}.<ref name="callan">{{cite arxiv|title=A combinatorial survey of identities for the double factorial|first=David|last=Callan|eprint=0906.1317|year=2009|ref=harv|class=math.CO}}</ref> That is,
:<math>n!! = \prod_{k=0}^{\left\lceil\frac{n}{2}\right\rceil - 1} (n-2k) = n (n-2) (n-4) \cdots </math>

(A consequence of this definition is that {{math|0!! {{=}} 1}}, as an [[empty product]].)

Therefore, for even {{mvar|n}} the double factorial is
:<math>n!! = \prod_{k=1}^\frac{n}{2} (2k) = n(n-2)(n-4)\cdots 4\cdot 2 \,,</math>
and for odd {{mvar|n}} it is
:<math>n!! = \prod_{k=1}^\frac{n+1}{2} (2k-1) = n(n-2)(n-4)\cdots 3\cdot 1 \,.</math>

For example, {{math|9!! {{=}} 9 × 7 × 5 × 3 × 1 {{=}} 945}}.

The double factorial should not be confused with the [[factorial]] function iterated twice, which is written as {{math|(''n''!)!}} and not {{math|''n''!!}}.

The sequence of double factorials for even {{mvar|n}} = {{math|0, 2, 4, 6, 8,...}} starts as
: 1, 2, 8, 48, 384, 3840, 46080, 645120,... {{OEIS|id=A000165}}
The sequence of double factorials for odd {{mvar|n}} = {{math|1, 3, 5, 7, 9,...}} starts as
: 1, 3, 15, 105, 945, 10395, 135135,... {{OEIS|id=A001147}}

<!--PREVIOUS TEXT, RETAINED IN CASE OF DISAGREEMENT:
In [[mathematics]], the product of all the odd integers up to some odd positive integer ''n'' is called the '''double factorial''', '''odd factorial''', or '''semifactorial''' of ''n'', and denoted by ''n''<nowiki>!!</nowiki>.<ref name="callan"/> That is,
:<math>(2k-1)!! = \prod_{i=1}^k (2i-1).</math>
For example, 9!!&nbsp;=&nbsp;1&nbsp;×&nbsp;3&nbsp;×&nbsp;5&nbsp;×&nbsp;7&nbsp;×&nbsp;9 =&nbsp;945.

The direct formula is of course defined only for odd numbers.  For odd ''n'' the double factorial may be expressed as
:<math>n!! = \prod_{i=1}^{(n+1)/2} (2i-1).</math>

The sequence of double factorials for odd ''n''&nbsp;=&nbsp;1,&nbsp;3,&nbsp;5,&nbsp;7,&nbsp;...  starts as
: 1, 3, 15, 105, 945, 10395, 135135, .... {{OEIS|id=A001147}}
-->
{{harvtxt|Meserve|1948}}<ref name="meserve">{{cite journal
 | last = Meserve | first = B. E.
 | doi = 10.2307/2306136
 | issue = 7
 | journal = [[The American Mathematical Monthly]]
 | mr = 1527019
 | pages = 425–426
 | title = Classroom Notes: Double Factorials
 | volume = 55
 | year = 1948|ref=harv| jstor = 2306136
 }}</ref> (possibly the earliest publication to use double factorial notation)<ref name="gq12"/> states that the double factorial was originally introduced in order to simplify the expression of certain [[List of integrals of trigonometric functions|trigonometric integrals]] arising in the derivation of the [[Wallis product]]. Double factorials also arise in expressing the volume of a [[n-sphere|hypersphere]], and they have many applications in [[enumerative combinatorics]].<ref name="callan"/><ref name="dm93"/> They occur in [[Student's t-distribution|Student's {{mvar|t}}-distribution]] (1908), though [[William Sealy Gosset|Gosset]] did not use the double exclamation point notation.

The term '''odd factorial''' is sometimes used for the double factorial of an odd number.<ref>{{cite journal
 | last1 = Henderson | first1 = Daniel J.
 | last2 = Parmeter | first2 = Christopher F.
 | doi = 10.1016/j.spl.2012.03.013
 | issue = 7
 | journal = Statistics & Probability Letters
 | mr = 2929790
 | pages = 1383–1387
 | title = Canonical higher-order kernels for density derivative estimation
 | volume = 82
 | year = 2012|ref=harv}}</ref>
<ref>{{cite journal
 | last = Nielsen | first = B.
 | doi = 10.1093/biomet/86.2.279
 | issue = 2
 | journal = Biometrika
 | mr = 1705359
 | pages = 279–288
 | title = The likelihood-ratio test for rank in bivariate canonical correlation analysis
 | volume = 86
 | year = 1999|ref=harv}}</ref>
<!--
The '''odd factorial''' of ''n'' is defined to be the product of all odd positive integers up to ''n'', that is, ''n''!! if ''n'' is odd and (''n''−1)!! if ''n'' is even.
-->

==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 {{math|''n''!}}, and it is much smaller than the iterated factorial {{math|(''n''!)!}}.

The factorial of a non-zero {{mvar|n}} may be written as the product of two double factorials:
:<math>n! = n!! \cdot (n-1)!!\,,</math>
and therefore
:<math>n!! = \frac{n!}{(n-1)!!} = \frac{(n+1)!}{(n+1)!!}\,,</math>
where the denominator cancels the unwanted factors in the numerator.  (The last form also applies when {{math|1=''n'' = 0}}.)

For an even positive integer {{math|''n'' {{=}} 2''k''}}, {{math|''k'' ≥ 0}}, the double factorial may be expressed as
:<math>n!! = 2^k k!\,.</math>

Combining these two, For odd {{math|''n'' {{=}} 2''k'' − 1}}, {{math|''k'' ≥ 1}}, the double factorial may be written
:<math>n!! = \frac{(2k)!}{2^k k!} = \frac{(2k-1)!}{2^{k-1} (k-1)!}\,.</math>

For an odd positive integer {{math|''n'' {{=}} 2''k'' − 1}}, {{math|''k'' ≥ 1}}, the double factorial may be expressed in terms of [[Permutation#Counting sequences without repetition|{{math|''k''}}-permutations of {{math|2''k''}}]] as<ref name="callan"/><ref name="gq12">{{cite journal
 | last1 = Gould | first1 = Henry
 | last2 = Quaintance | first2 = Jocelyn
 | doi = 10.4169/math.mag.85.3.177
 | issue = 3
 | journal = [[Mathematics Magazine]]
 | mr = 2924154
 | pages = 177–192
 | title = Double fun with double factorials
 | volume = 85
 | year = 2012|ref=harv}}</ref>
:<math>(2k-1)!! = \frac {_{2k}P_k} {2^k} = \frac {(2k)^{\underline k}} {2^k}\,.</math>

==Applications in enumerative combinatorics==
[[File:Unordered binary trees with 4 leaves.svg|thumb|300px|The fifteen different [[rooted binary tree]]s (with unordered children) on a set of four labeled leaves, illustrating {{math|15 {{=}} (2 × 4 − 3)!!}} (see article text).]]
Double factorials are motivated by the fact that they occur frequently in [[enumerative combinatorics]] and other settings. For instance, {{math|''n''!!}} for odd values of {{mvar|n}} counts
*[[Perfect matching]]s of the [[complete graph]] {{math|''K''<sub>''n'' + 1</sub>}} for odd {{mvar|n}}. In such a graph, any single vertex ''v'' has {{mvar|n}} 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''.<ref name="callan"/> Perfect matchings may be described in several other equivalent ways, including [[Involution (mathematics)|involutions]] without fixed points on a set of {{math|''n'' + 1}} items ([[permutations]] in which each cycle is a pair)<ref name="callan"/> or chord diagrams (sets of chords of a set of {{math|''n'' + 1}} points evenly spaced on a circle such that each point is the endpoint of exactly one chord, also called [[Richard Brauer|Brauer]] diagrams).<ref name="dm93"/><ref>{{cite book|title=Patterns in Permutations and Words|series=EATCS Monographs in Theoretical Computer Science |first=Sergey|last=Kitaev|publisher=Springer|year=2011|isbn=9783642173332|page=96|url=https://books.google.com/books?id=JgQHtgR5N60C&pg=PA96|ref=harv}}</ref><ref>{{cite journal
 | last1 = Dale | first1 = M. R. T.
 | last2 = Narayana | first2 = T. V.
 | doi = 10.1016/0378-3758(86)90161-8
 | issue = 2
 | journal = Journal of Statistical Planning and Inference
 | mr = 852528
 | pages = 245–249
 | title = A partition of Catalan permuted sequences with applications
 | volume = 14
 | year = 1986|ref=harv}}</ref> The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are instead given by the [[Telephone number (mathematics)|telephone number]]s, which may be expressed as a summation involving double factorials.<ref>{{cite journal
 | last1 = Tichy | first1 = Robert F.
 | last2 = Wagner | first2 = Stephan
 | doi = 10.1089/cmb.2005.12.1004
 | issue = 7
 | journal = [[Journal of Computational Biology]]
 | pages = 1004–1013
 | title = Extremal problems for topological indices in combinatorial chemistry
 | url = http://www.math.tugraz.at/fosp/pdfs/tugraz_main_0052.pdf
 | volume = 12
 | year = 2005|ref=harv}}</ref>
*[[Stirling permutation]]s, permutations of the [[multiset]] of numbers {{math|1, 1, 2, 2, ..., {{mvar|k}}, {{mvar|k}}}} in which each pair of equal numbers is separated only by larger numbers, where {{math|''k'' {{=}} {{sfrac|''n'' + 1|2}}}}. The two copies of {{mvar|k}} must be adjacent; removing them from the permutation leaves a permutation in which the maximum element is {{math|''k'' − 1}}, with {{mvar|n}} positions into which the adjacent pair of {{mvar|k}} values may be placed. From this recursive construction, a proof that the Stirling permutations are counted by the double permutations follows by induction.<ref name="callan"/> 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 {{math|2''k''}} positions of the permutation, so again the number of permutations may be counted by the double permutations.<ref name="dm93">{{cite journal
 | last1 = Dale | first1 = M. R. T.
 | last2 = Moon | first2 = J. W.
 | doi = 10.1016/0378-3758(93)90035-5
 | issue = 1
 | journal = Journal of Statistical Planning and Inference
 | mr = 1209991
 | pages = 75–87
 | title = The permuted analogues of three Catalan sets
 | volume = 34
 | year = 1993|ref=harv}}</ref>
*[[Heap (data structure)|Heap-ordered tree]]s, trees with {{math|''k'' + 1}} nodes labeled {{math|0, 1, 2, ... {{mvar|k}}}}, 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 technique|Euler tour]] of the tree (with doubled edges) gives a Stirling permutation, and every Stirling permutation represents a tree in this way.<ref name="callan"/><ref>{{cite conference
 | last = Janson | first = Svante | authorlink = Svante Janson
 | arxiv = 0803.1129
 | contribution = Plane recursive trees, Stirling permutations and an urn model
 | mr = 2508813
 | pages = 541–547
 | publisher = Assoc. Discrete Math. Theor. Comput. Sci., Nancy
 | series = Discrete Math. Theor. Comput. Sci. Proc., AI
 | title = Fifth Colloquium on Mathematics and Computer Science
 | year = 2008|ref=harv| bibcode = 2008arXiv0803.1129J}}</ref>
*[[Unrooted binary tree]]s with {{math|{{sfrac|''n'' + 5|2}}}} labeled leaves. Each such tree may be formed from a tree with one fewer leaf, by subdividing one of the {{mvar|n}} tree edges and making the new vertex be the parent of a new leaf.
*[[Rooted binary tree]]s with {{math|{{sfrac|''n'' + 3|2}}}} 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.<ref name="callan"/><ref name="dm93"/>

{{harvtxt|Callan|2009}} and {{harvtxt|Dale|Moon|1993}} list several additional objects with the same [[combinatorial class|counting sequence]], including "trapezoidal words" ([[numeral system|numerals]] in a [[mixed radix]] system with increasing odd radixes), height-labeled [[Dyck path]]s, 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 proof]]s that some of these objects are equinumerous, see {{harvtxt|Rubey|2008}} and {{harvtxt|Marsh|Martin|2011}}.<ref>{{cite conference
 | last = Rubey | first = Martin
 | contribution = Nestings of matchings and permutations and north steps in PDSAWs
 | mr = 2721495
 | pages = 691–704
 | publisher = Assoc. Discrete Math. Theor. Comput. Sci., Nancy
 | series = Discrete Math. Theor. Comput. Sci. Proc., AJ
 | title = 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)
 | year = 2008|ref=harv}}</ref><ref>{{cite journal
 | last1 = Marsh | first1 = Robert J.
 | last2 = Martin | first2 = Paul
 | arxiv = 0906.0912
 | doi = 10.1007/s10801-010-0252-6
 | issue = 3
 | journal = [[Journal of Algebraic Combinatorics]]
 | mr = 2772541
 | pages = 427–453
 | title = Tiling bijections between paths and Brauer diagrams
 | volume = 33
 | year = 2011|ref=harv}}</ref>

The even double factorials give the numbers of elements of the [[hyperoctahedral group]]s (signed permutations or symmetries of a [[hypercube]])

==Extensions==

===Negative arguments===
The ordinary factorial, when extended to the [[gamma function]], has a [[Pole (complex analysis)|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]]
:<math>n!! = n \times (n-2)!!</math>
to give
:<math>n!! = \frac{(n+2)!!}{n+2}\,.</math>
Using this inverted recurrence, −1!!&nbsp;=&nbsp;1, −3!!&nbsp;=&nbsp;−1, and −5!!&nbsp;=&nbsp;{{sfrac|1|3}}; negative odd numbers with greater magnitude have fractional double factorials.<ref name="callan"/>  In particular, this gives, when {{mvar|n}} is an odd number,
:<math>(-n)!! \times n!! = (-1)^\frac{n-1}{2} \times n\,.</math>
<!-- RETAINED IN CASE OF DISAGREEMENT
===Even arguments===
Sometimes ''n''<nowiki>!!</nowiki> is defined for non-negative even integers as well. One choice is a definition similar to the one for odd values:<ref name="callan"/><ref name="meserve"/>
:<math>(2k)!!= \prod_{i=1}^k (2i) = 2^k k!.</math>
One consequence of this definition is that (as an [[empty product]])
:<math>0!! = 1.</math>

The sequence of double factorials for ''n''&nbsp;=&nbsp;0,&nbsp;2,&nbsp;4,&nbsp;6,&nbsp;8,&nbsp;...  starts as
: 1, 2, 8, 48, 384, 3840, 46080, 645120, .... {{OEIS|id=A000165}}

These double factorials for even and odd integers share the following property for all positive integers n:
: <math>n! = n!! \times (n-1)!!.</math>

However, this definition does not match the expression for odd arguments of the double factorial in terms of the ordinary factorial, and is also inconsistent with the extension of the definition of <math>n!!</math> to complex numbers <math>n</math> that is achieved via the [[Gamma function]] below.
-->

===Complex arguments===
Disregarding the above definition of {{math|''n''!!}} for even values of&nbsp;{{mvar|n}}, the double factorial for odd integers can be extended to most real and complex numbers {{mvar|z}} by noting that when {{mvar|z}} is a positive odd integer then<ref>{{cite book|title=Mathematical Methods: For Students of Physics and Related Fields|series=[[Undergraduate Texts in Mathematics]]|first=Sadri|last=Hassani|publisher=Springer|year=2000|isbn=9780387989587|page=266|url=https://books.google.com/books?id=dxSOzeLMij4C&pg=PA266|ref=harv}}</ref><ref>{{cite web|title=Double factorial: Specific values (formula 06.02.03.0005) |publisher=Wolfram Research|date=2001-10-29 |url=http://functions.wolfram.com/06.02.03.0005 |accessdate=2013-03-23}}</ref>

:<math>\begin{align} z!! &= z(z-2)\cdots 3 \cdot 1 \\
&= 2^\frac{z-1}{2}\left(\frac{z}{2}\right)\left(\frac{z-2}{2}\right)\cdots \left(\frac{3}{2}\right) \\
&= 2^\frac{z-1}{2} \frac{\Gamma\left(\frac{z}{2}+1\right)}{\Gamma\left(\frac{1}{2}+1\right)} \\
&= \sqrt{\frac{2^{z+1}}{\pi}} \Gamma\left(\frac{z}{2}+1\right) = \left(\frac{z}{2}\right)!\sqrt{\frac{2^{z+1}}{\pi}} \,.\end{align}</math>

From this one can derive an alternative definition of {{math|''z''!!}} for non-negative even integer values of&nbsp;{{mvar|z}}:
:<math>(2k)!!= \sqrt{ \frac{2}{\pi} } \prod_{i=1}^k (2i) = 2^k k! \sqrt{ \frac{2}{\pi} } \,,</math>
with the value for 0!! in this case being
:<math>0!! = \sqrt{ \frac{2}{\pi} } \approx 0.797\,884\,5608\dots \,.</math>

The expression found for {{math|''z''!!}} is defined for all complex numbers except the negative even integers. Using it as the definition, the [[Volume of an n-ball|volume]] of an {{mvar|n}}-[[dimension]]al [[hypersphere]] of radius {{mvar|R}} can be expressed as<ref>{{cite journal|title=Some dimension problems in molecular databases|first=Paul G.|last=Mezey|year=2009|journal=Journal of Mathematical Chemistry|volume=45|issue=1|pages=1–6|doi=10.1007/s10910-008-9365-8|ref=harv}}</ref>

:<math>V_n=\frac{2 (2\pi)^\frac{n-1}{2}}{n!!} R^n\,.</math>

==Additional identities==
For integer values of {{mvar|n}},
<!-- FORMER TEXT Using the extension of the double factorial to even arguments, for even values of ''n'', -->
:<math>\int_{0}^\frac{\pi}{2}\sin^n x\,dx=\int_{0}^\frac{\pi}{2}\cos^n x\,dx=\frac{(n-1)!!}{n!!}\times
\begin{cases}1 & \text{if } n \text{ is odd} \\ \frac{\pi}{2} & \text{if } n \text{ is even.}\end{cases}</math>
Using instead the extension of the double factorial of odd numbers to complex numbers, the formula is
:<math>\int_{0}^\frac{\pi}{2}\sin^n x\,dx=\int_{0}^\frac{\pi}{2}\cos^n x\,dx=\frac{(n-1)!!}{n!!} \sqrt{\frac{\pi}{2}}\,.</math>

Double factorials can also be used to evaluate integrals of more complicated trigonometric polynomials.<ref name="meserve"/><ref>{{cite journal
 | last1 = Dassios | first1 = George
 | last2 = Kiriaki | first2 = Kiriakie
 | issue = A
 | journal = Bulletin de la Société Mathématique de Grèce
 | mr = 935868
 | pages = 40–43
 | title = A useful application of Gauss theorem
 | volume = 28
 | year = 1987|ref=harv}}</ref>

Double factorials of odd numbers are related to the [[gamma function]] by the identity:

:<math>(2n-1)!! = 2^n \cdot \frac{\Gamma\left(\frac{1}{2} + n\right)} {\sqrt{\pi}} = (-2)^n \cdot \frac{\sqrt{\pi}} { \Gamma\left(\frac{1}{2} - n\right)}\,.</math>

Some additional identities involving double factorials of odd numbers are:<ref name="callan"/>

:<math>\begin{align}
(2n-1)!! &= \sum_{k=1}^{n-1} \binom{n}{k+1} (2k-1)!! (2n-2k-3)!!\,, \\
(2n-1)!! &= \sum_{k=0}^{n} \binom{2n-k-1}{k-1} \frac{(2k-1)(2n-k+1)}{k+1}(2n-2k-3)!!\,, \\
(2n-1)!! &= \sum_{k=1}^{n} \frac{(n-1)!}{(k-1)!} k(2k-3)!!\,.
\end{align}</math>

An approximation for the ratio of the double factorial of two consecutive integers is
:<math> \frac{(2n)!!}{(2n-1)!!} \approx \sqrt{\pi n}. </math>
This approximation gets more accurate as {{mvar|n}} increases.

==Generalizations==

===Definitions===

In the same way that the double factorial generalizes the notion of the [[Factorial|single factorial]], the following definition of the integer-valued multiple factorial functions ([[Factorial#Multifactorials|multifactorials]]), or {{mvar|α}}-factorial functions, extends the notion of the double factorial function for {{math|''α'' ∈ ℤ<sup>+</sup>}}:

:<math>
n!_{(\alpha)} = \begin{cases} 
     n \cdot (n-\alpha)!_{(\alpha)} & \text{ if } n > 0 \,; \\ 
     1 & \text{ if } -\alpha < n \leq 0 \,; \\ 
     0 & \text{ otherwise. } 
\end{cases}
</math>

===Alternative extension of the multifactorial===
Alternatively, the multifactorial {{math|''n''!<sub>(''α'')</sub>}} can be extended to most real and complex numbers {{math|''n''}} by noting that when {{math|''n''}} is one more than a positive multiple of {{math|''α''}} then

:<math>\begin{align} n!_{(\alpha)} &= n(n-\alpha)\cdots (\alpha+1) \\
&= \alpha^\frac{n-1}{\alpha}\left(\frac{n}{\alpha}\right)\left(\frac{n-\alpha}{\alpha}\right)\cdots \left(\frac{\alpha+1}{\alpha}\right) \\
&= \alpha^\frac{n-1}{\alpha} \frac{\Gamma\left(\frac{n}{\alpha}+1\right)}{\Gamma\left(\frac{1}{\alpha}+1\right)}\,.
\end{align}</math>

This last expression is defined much more broadly than the original.  In the same way that {{math|''n''!}} is not defined for negative integers, and {{math|''n''!!}} is not defined for negative even integers, {{math|''n''!<sub>(''α'')</sub>}} is not defined for negative multiples of {{math|''α''}}.  However, it is defined for all other complex numbers. This definition is consistent with the earlier definition only for those integers {{math|''n''}} satisfying&nbsp;{{math|''n'' ≡ 1 mod ''α''}}.

In addition to extending {{math|''n''!<sub>(''α'')</sub>}} to most complex numbers&nbsp;{{math|''n''}}, this definition has the feature of working for all positive real values of&nbsp;{{math|''α''}}. Furthermore, when {{math|1=''α'' = 1}}, this definition is mathematically equivalent to the {{math|Π(''n'')}} function, described above. Also, when {{math|1=''α'' = 2}}, this definition is mathematically equivalent to the [[Double factorial#Complex arguments|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 {{math|''α'' > 0}} by the following triangular recurrence relation:

:<math>\left[\begin{matrix} n \\ k \end{matrix} \right]_{\alpha} 
     = (\alpha n+1-2\alpha) \left[\begin{matrix} n-1 \\ k \end{matrix} \right]_{\alpha} + \left[\begin{matrix} n-1 \\ k-1 \end{matrix} \right]_{\alpha} + \delta_{n,0} \delta_{k,0}\,. 
</math>

These generalized ''{{mvar|α}}-factorial coefficients'' then generate the distinct symbolic polynomial products defining the multiple factorial, or {{mvar|α}}-factorial functions, {{math|(''x'' − 1)!<sub>(''α'')</sub>}}, as

:<math>
\begin{align} 
(x-1|\alpha)^{\underline{n}} & := \prod_{i=0}^{n-1} \left(x-1-i\alpha\right) = (x-1)(x-1-\alpha)\cdots\bigl(x-1-(n-1)\alpha\bigr) \\ 
& = \sum_{k=0}^n \left[\begin{matrix} n \\ k \end{matrix} \right] (-\alpha)^{n-k} (x-1)^k \\ 
& = \sum_{k=1}^n \left[\begin{matrix} n \\ k \end{matrix} \right]_{\alpha} (-1)^{n-k} x^{k-1}\,.
\end{align} 
</math>

The distinct polynomial expansions in the previous equations actually define the {{mvar|α}}-factorial products for multiple distinct cases of the least residues {{math|''x'' ≡ ''n''<sub>0</sub> mod ''α''}} for {{math|''n''<sub>0</sub> ∈ {0, 1, 2, ..., ''α'' − 1<nowiki>}</nowiki>}}.

The generalized {{mvar|α}}-factorial polynomials, {{math|''σ''{{su|b=''n''|p=(''α'')}}(''x'')}} where {{math|''σ''{{su|b=''n''|p=(1)}}(''x'') ≡ ''σ''<sub>''n''</sub>(''x'')}}, which generalize the [[Stirling polynomial#Stirling convolution polynomials|Stirling convolution polynomials]] from the single factorial case to the multifactorial cases, are defined by

:<math>\sigma_n^{(\alpha)}(x) := \left[\begin{matrix} x \\ x-n \end{matrix} \right]_{(\alpha)} \frac{(x-n-1)!}{x!}</math>

for {{math|0 ≤ ''n'' ≤ ''x''}}. These polynomials have a particularly nice closed-form [[ordinary generating function]] given by

:<math>\sum_{n \geq 0} x \cdot \sigma_n^{(\alpha)}(x) z^n = e^{(1-\alpha)z} \left(\frac{\alpha z e^{\alpha z}}{e^{\alpha z}-1}\right)^x\,. </math>

Other combinatorial properties and expansions of these generalized {{mvar|α}}-factorial triangles and polynomial sequences are considered in {{harvtxt|Schmidt|2010}}.<ref>{{cite journal|last1=Schmidt|first1=Maxie D.|title=Generalized ''j''-Factorial Functions, Polynomials, and Applications|journal=J. Integer Seq.|date=2010|volume=13|url=https://cs.uwaterloo.ca/journals/JIS/VOL13/Schmidt/multifact.html|ref=harv}}</ref>

===Exact finite sums involving multiple factorial functions===

Suppose that {{math|''n'' ≥ 1}} and {{math|''α'' ≥ 2}} are integer-valued. Then we can expand the next single finite sums involving the multifactorial, or {{mvar|α}}-factorial functions, {{math|(''αn'' − 1)!<sub>(''α'')</sub>}}, in terms of the [[Pochhammer symbol]] and the generalized, rational-valued [[binomial coefficients]] as

:<math>
\begin{align} 
(\alpha n-1)!_{(\alpha)} & = \sum_{k=0}^{n-1} \binom{n-1}{k+1} (-1)^k \times \left(\frac{1}{\alpha}\right)_{-(k+1)} \left(\frac{1}{\alpha}-n\right)_{k+1} \times \bigl(\alpha(k+1)-1\bigr)!_{(\alpha)} \bigl(\alpha(n-k-1)-1\bigr)!_{(\alpha)} \\ 
& = \sum_{k=0}^{n-1} \binom{n-1}{k+1} (-1)^k \times \binom{\frac{1}{\alpha}+k-n}{k+1} \binom{\frac{1}{\alpha}-1}{k+1} \times \bigl(\alpha(k+1)-1\bigr)!_{(\alpha)} \bigl(\alpha(n-k-1)-1\bigr)!_{(\alpha)}\,, 
\end{align} 
</math>

and moreover, we similarly have double sum expansions of these functions given by

:<math> 
\begin{align}
(\alpha n-1)!_{(\alpha)} & = \sum_{k=0}^{n-1} \sum_{i=0}^{k+1} \binom{n-1}{k+1} \binom{k+1}{i} (-1)^k \alpha^{k+1-i} (\alpha i-1)!_{(\alpha)} \bigl(\alpha(n-1-k)-1\bigr)!_{(\alpha)} \times (n-1-k)_{k+1-i} \\ 
& = \sum_{k=0}^{n-1} \sum_{i=0}^{k+1} \binom{n-1}{k+1} \binom{k+1}{i} \binom{n-1-i}{k+1-i} (-1)^k \alpha^{k+1-i} (\alpha i-1)!_{(\alpha)} \bigl(\alpha(n-1-k)-1\bigr)!_{(\alpha)} \times (k+1-i)!.
\end{align} 
</math>

The first two sums above are similar in form to a known ''non-round'' combinatorial identity for the double factorial function when {{math|1=''α'' := 2}} given by {{harvtxt|Callan|2009}}.

:<math>(2n-1)!! = \sum_{k=0}^{n-1} \binom{n}{k+1} (2k-1)!! (2n-2k-3)!!.</math>

Additional finite sum expansions of congruences for the {{mvar|α}}-factorial functions, {{math|(''αn'' − ''d'')!<sub>(''α'')</sub>}}, modulo any prescribed integer {{math|''h'' ≥ 2}} for any {{math|0 ≤ ''d'' < ''α''}} are given by {{harvtxt|Schmidt|2017}}.<ref>{{cite arxiv|last1=Schmidt|first1=Maxie D.|title=New Congruences and Finite Difference Equations for Generalized Factorial Functions|eprint=1701.04741|year=2017|ref=harv|class=math.CO}}</ref>

==References==
{{reflist}}

==External links==
*{{mathworld|id=DoubleFactorial|title=Double Factorial}}

[[Category:Integer sequences]]
[[Category:Enumerative combinatorics]]
[[Category:Factorial and binomial topics]]

[[fr:Analogues de la factorielle#Multifactorielles]]