Editing Double factorial

{{Short description|Mathematical function}}
{{hatnote|The double factorial <math>n!!</math> is not the same as applying the [[factorial]] function twice <math>(n!)!</math>.}}[[File:Chord diagrams K6 matchings.svg|thumb|360px|The fifteen different [[Chord diagram (mathematics)|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''' of a number {{mvar|n}}, denoted by {{math|''n''‼}}, is the product of all the [[positive integer]]s 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|class=math.CO}}</ref> That is,

<math display="block">n!! = \prod_{k=0}^{\left\lceil\frac{n}{2}\right\rceil - 1} (n-2k) = n (n-2) (n-4) \cdots.</math>

Restated, this says that for even {{mvar|n}}, the double factorial<ref>Some authors define the double factorial differently for even numbers; see {{slink|Double factorial|complex arguments}} below.</ref> is
<math display="block">n!! = \prod_{k=1}^\frac{n}{2} (2k) = n(n-2)(n-4)\cdots 4\cdot 2 \,,</math>
while for odd {{mvar|n}} it is
<math display="block">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 zero double factorial {{math|0‼ {{=}} 1}} as an [[empty product]].<ref name=":0">{{Cite web|last=Weisstein|first=Eric W.|title=Double Factorial|url=https://mathworld.wolfram.com/DoubleFactorial.html|access-date=2020-09-10|website=mathworld.wolfram.com|language=en}}</ref><ref>{{Cite web|title=Double Factorials and Multifactorials {{!}} Brilliant Math & Science Wiki|url=https://brilliant.org/wiki/double-factorials-and-multifactorials/|access-date=2020-09-10|website=brilliant.org|language=en-us}}</ref>

The [[sequence]] of double factorials for even {{mvar|n}} = {{math|0, 2, 4, 6, 8,...}} starts as
{{block indent|{{math|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
{{block indent|{{math|1, 3, 15, 105, 945, 10395, 135135, ...}} {{OEIS|id=A001147}} }}

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><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>
<!--
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.
-->

The term '''semifactorial''' is also used by [[Donald Knuth|Knuth]] as a synonym of double factorial.<ref>{{Cite book |last=Knuth |first=Donald Ervin |authorlink=Donald Knuth |title=The art of computer programming. volume 4B part 2: Combinatorial algorithms |date=2023 |publisher=Addison-Wesley |isbn=978-0-201-03806-4 |location=Boston Munich}}</ref>

==History and usage==
In a 1902 paper, the physicist [[Arthur Schuster]] wrote:<ref>{{cite journal
 | last = Schuster | first = Arthur
 | doi = 10.1098/rspl.1902.0068 | doi-access = free
 | journal = Proceedings of the Royal Society of London
 | jstor = 116355
 | pages = 97–101
 | title = On some definite integrals and a new method of reducing a function of spherical co-ordinates to a series of spherical harmonics
 | volume = 71
 | year = 1902| issue = 467–476
 }} See in particular p. 99.</ref>
{{quote|1=The symbolical representation of the results of this paper is much facilitated by the introduction of a separate symbol for the product of alternate factors, <math>n \cdot n-2 \cdot n-4 \cdots 1</math>, if <math>n</math> be odd, or <math>n \cdot n-2 \cdots 2</math> if <math>n</math> be odd [sic]. I propose to write <math>n!!</math> for such products, and if a name be required for the product to call it the "alternate factorial" or the "double factorial".}}

{{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| jstor = 2306136
 }}</ref> states that the double factorial was originally introduced in order to simplify the expression of certain [[List of integrals of trigonometric functions|trigonometric integrals]] that arise in the derivation of the [[Wallis product]]. Double factorials also arise in expressing the volume of a [[Ball (mathematics)|hyperball]] and surface area 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.

==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 positive {{mvar|n}} may be written as the product of two double factorials:<ref name=":0" />
<math display="block">n! = n!! \cdot (n-1)!!\,,</math>
and therefore
<math display="block">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 non-negative integer {{math|''n'' {{=}} 2''k''}} with {{math|''k'' ≥ 0}}, the double factorial may be expressed as
<math display="block">(2k)!! = 2^k k!\,.</math>

For odd {{math|''n'' {{=}} 2''k'' − 1}} with {{math|''k'' ≥ 1}}, combining the two previous formulas yields
<math display="block">(2k-1)!! = \frac{(2k)!}{2^k k!} = \frac{(2k-1)!}{2^{k-1} (k-1)!}\,.</math>

For an odd positive integer {{math|''n'' {{=}} 2''k'' − 1}} with {{math|''k'' ≥ 1}}, the double factorial may be expressed in terms of [[Permutation#Permutations without repetitions|{{mvar|k}}-permutations of {{math|2''k''}}]]<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| s2cid = 117120280
 }}</ref> or a [[Falling and rising factorials|falling factorial]] as
<math display="block">(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 diagram (mathematics)|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| author-link = Sergey Kitaev|publisher=Springer|year=2011|isbn=9783642173332|page=96|url=https://books.google.com/books?id=JgQHtgR5N60C&pg=PA96}}</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> 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| pmid = 16201918}}</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>
*[[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 | author-link = 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| 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><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| s2cid = 7264692
 }}</ref>

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

==Asymptotics==
[[Stirling's approximation]] for the factorial can be used to derive an [[Asymptotic analysis|asymptotic equivalent]] for the double factorial.  In particular, since <math>n! \sim \sqrt{2 \pi n}\left(\frac{n}{e}\right)^n,</math> one has as <math>n</math> tends to infinity that

<math display=block>n!! \sim \begin{cases}
\displaystyle \sqrt{\pi n}\left(\frac{n}{e}\right)^{n/2} & \text{if } n \text{ is even}, \\[5pt]
\displaystyle \sqrt{2 n}\left(\frac{n}{e}\right)^{n/2} & \text{if } n \text{ is odd}.
\end{cases}</math>

==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 display="block">n!! = n \times (n-2)!!</math>
to give
<math display="block">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, when {{mvar|n}} is an odd number, this gives
<math display="block">(-n)!! \times n!! = (-1)^\frac{n-1}{2} \times n\,.</math>

===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><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 |access-date=2013-03-23}}</ref>

<math display="block">\begin{align}
z!!
&= z(z-2)\cdots 5 \cdot 3 \\[3mu]
&= 2^\frac{z-1}{2}\left(\frac z2\right)\left(\frac{z-2}2\right)\cdots \left(\frac{5}{2}\right) \left(\frac{3}{2}\right) \\[5mu]
&= 2^\frac{z-1}{2} \frac{\Gamma\left(\tfrac z2+1\right)}{\Gamma\left(\tfrac12+1\right)} \\[5mu]
&= \sqrt{\frac{2}{\pi}} 2^\frac{z}{2} \Gamma\left(\tfrac z2+1\right) \,,\end{align}</math>
where <math>\Gamma(z)</math> is the [[gamma function]].

The final expression is defined for all complex numbers except the negative even integers and satisfies {{math|1=(''z'' + 2)!! = (''z'' + 2) · ''z''!!}} 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, <math display=inline>n!! \sim \sqrt{2 n^{n+1} e^{-n}}\,.</math>

The generalized formula <math>\sqrt{\frac{2}{\pi}} 2^\frac{z}{2} \Gamma\left(\tfrac z2+1\right)</math> does not agree with the previous product formula for {{math|''z''!!}} for non-negative ''even'' integer values of&nbsp;{{mvar|z}}.  Instead, this generalized formula implies the following alternative:
<math display="block">(2k)!! = \sqrt{\frac{2}{\pi}} 2^k \Gamma\left(k+1\right) = \sqrt{ \frac{2}{\pi} } \prod_{i=1}^k (2i) \,,</math>
with the value for 0!! in this case being
{{startplainlist|indent=1}}
* <math>0!! = \sqrt{ \frac{2}{\pi} } \approx 0.797\,884\,5608\dots</math> {{OEIS|id=A076668}}.
{{endplainlist}}

Using this generalized formula 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|s2cid=120103389}}</ref>

<math display="block">V_n=\frac{2 \left(2\pi\right)^\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 display="block">\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 display="block">\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 | author1-link = George Dassios
 | 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>

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

<math display="block">(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 display="block">\begin{align}
(2n-1)!! &= \sum_{k=0}^{n-1} \binom{n}{k+1} (2k-1)!! (2n-2k-3)!! \\
  &= \sum_{k=1}^{n} \binom{n}{k} (2k-3)!! (2(n-k)-1)!! \\
  &= \sum_{k=0}^{n} \binom{2n-k-1}{k-1} \frac{(2k-1)(2n-k+1)}{k+1}(2n-2k-3)!! \\
  &= \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 display="block"> \frac{(2n)!!}{(2n-1)!!} \approx \sqrt{\pi n}. </math>
This approximation gets more accurate as {{mvar|n}} increases, which can be seen as a result of the [[Wallis%27_integrals#Deducing the Double Factorial Ratio | Wallis Integral]].

==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 (multifactorials), or {{mvar|α}}-factorial functions, extends the notion of the double factorial function for positive integers <math>\alpha</math>:

<math display="block">
n!_{(\alpha)} = \begin{cases}
     n \cdot (n-\alpha)!_{(\alpha)} & \text{ if } n > \alpha \,; \\
     n & \text{ if } 1 \leq n \leq \alpha \,; \text{and} \\
     (n+\alpha)!_{(\alpha)} / (n+\alpha) & \text{ if } n \leq 0 \text{ and is not a negative multiple of } \alpha \,;
\end{cases}
</math>

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

<math display="block">\begin{align} z!_{(\alpha)} &= z(z-\alpha)\cdots (\alpha+1) \\
&= \alpha^\frac{z-1}{\alpha}\left(\frac{z}{\alpha}\right)\left(\frac{z-\alpha}{\alpha}\right)\cdots \left(\frac{\alpha+1}{\alpha}\right) \\
&= \alpha^\frac{z-1}{\alpha} \frac{\Gamma\left(\frac{z}{\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|''z''!}} is not defined for negative integers, and {{math|''z''‼}} is not defined for negative even integers, {{math|''z''!<sub>(''α'')</sub>}} is not defined for negative multiples of {{mvar|α}}.  However, it is defined and satisfies {{math|1=(''z''+''α'')!<sub>(''α'')</sub> = (''z''+''α'')·''z''!<sub>(''α'')</sub>}} for all other complex numbers&nbsp;{{mvar|z}}. This definition is consistent with the earlier definition only for those integers {{mvar|z}} satisfying&nbsp;{{math|''z'' ≡ 1 mod ''α''}}.

In addition to extending {{math|''z''!<sub>(''α'')</sub>}} to most complex numbers&nbsp;{{mvar|z}}, this definition has the feature of working for all positive real values of&nbsp;{{mvar|α}}. Furthermore, when {{math|1=''α'' = 1}}, this definition is mathematically equivalent to the {{math|Π(''z'')}} function, described above. Also, when {{math|1=''α'' = 2}}, this definition is mathematically equivalent to the [[#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 display="block">\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 display="block">
\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 display="block">\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 display="block">\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>

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

Similar identities can be obtained via context-free grammars.<ref>{{Cite journal|last1=Triana |first1=Juan |last2=De Castro |first2=Rodrigo |year=2019 |title=The formal derivative operator and multifactorial numbers|journal=Revista Colombiana de Matemáticas|volume=53 |issue=2 |pages=125–137 |doi=10.15446/recolma.v53n2.85522 |issn=0034-7426 |doi-access=free }}</ref> 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|2018}}.<ref>{{cite journal | last = Schmidt | first = Maxie D. | arxiv = 1701.04741 | journal = Integers | mr = 3862591 | pages = A78:1–A78:34 | title = New congruences and finite difference equations for generalized factorial functions | url = https://math.colgate.edu/~integers/s78/s78.pdf | volume = 18 | year = 2018}}</ref>

==References==
{{reflist}}
[[Category:Integer sequences]]
[[Category:Enumerative combinatorics]]
[[Category:Factorial and binomial topics]]

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