double factorial

In a 1902 paper, the physicist Arthur Schuster wrote:{{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.

{{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, $n\; \backslash cdot\; n-2\; \backslash cdot\; n-4\; \backslash cdots\; 1$, if $n$ be odd, or $n\; \backslash cdot\; n-2\; \backslash cdots\; 2$ if $n$ be odd [sic]. I propose to write $n!!$ 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}}{{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

}} 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 hyperball and surface area of a hypersphere, and they have many applications in enumerative combinatorics. They occur in Student's t-distribution (1908), though Gosset did not use the double exclamation point notation.

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:

$$n!\; =\; n!!\; \backslash cdot\; (n-1)!!\backslash ,,$$

and therefore

$$n!!\; =\; \backslash frac\{n!\}\{(n-1)!!\}\; =\; \backslash frac\{(n+1)!\}\{(n+1)!!\}\backslash ,,$$

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 {{=}} 2k}} with {{math|k ≥ 0}}, the double factorial may be expressed as

$$(2k)!!\; =\; 2^k\; k!\backslash ,.$$

For odd {{math|n {{=}} 2k − 1}} with {{math|k ≥ 1}}, combining the two previous formulas yields

$$(2k-1)!!\; =\; \backslash frac\{(2k)!\}\{2^k\; k!\}\; =\; \backslash frac\{(2k-1)!\}\{2^\{k-1\}\; (k-1)!\}\backslash ,.$$

For an odd positive integer {{math|n {{=}} 2k − 1}} with {{math|k ≥ 1}}, the double factorial may be expressed in terms of Permutation#Permutations without repetitions{{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

}} or a falling factorial as

$$(2k-1)!!\; =\; \backslash frac\; \{\_\{2k\}P\_k\}\; \{2^k\}\; =\; \backslash frac\; \{(2k)^\{\backslash underline\; k\}\}\; \{2^k\}\backslash ,.$$

File:Unordered binary trees with 4 leaves.svgs (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 matchings of the complete graph {{math|K_{n + 1}}} 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. Perfect matchings may be described in several other equivalent ways, including involutions without fixed points on a set of {{math|n + 1}} items (permutations in which each cycle is a pair) 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 Brauer diagrams).{{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}}{{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}} 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.{{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}}

• Stirling permutations, 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. 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|2k}} positions of the permutation, so again the number of permutations may be counted by the double permutations.{{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}}

• Heap-ordered trees, 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 of the tree (with doubled edges) gives a Stirling permutation, and every Stirling permutation represents a tree in this way.{{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}}

• Unrooted binary trees 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 trees 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.

{{harvtxt|Callan|2009}} and {{harvtxt|Dale|Moon|1993}} 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 {{harvtxt|Rubey|2008}} and {{harvtxt|Marsh|Martin|2011}}.{{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}}{{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

}}

The even double factorials give the numbers of elements of the hyperoctahedral groups (signed permutations or symmetries of a hypercube)

Stirling's approximation for the factorial can be used to derive an asymptotic equivalent for the double factorial. In particular, since $n!\; \backslash sim\; \backslash sqrt\{2\; \backslash pi\; n\}\backslash left(\backslash frac\{n\}\{e\}\backslash right)^n,$ one has as $n$ tends to infinity that

$$n!!\; \backslash sim\; \backslash begin\{cases\}$$

\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}

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

$$n!!\; =\; n\; \backslash times\; (n-2)!!$$

to give

$$n!!\; =\; \backslash frac\{(n+2)!!\}\{n+2\}\backslash ,.$$

Using this inverted recurrence, (−1)!! = 1, (−3)!! = −1, and (−5)!! = {{sfrac|1|3}}; negative odd numbers with greater magnitude have fractional double factorials. In particular, when {{mvar|n}} is an odd number, this gives

$$(-n)!!\; \backslash times\; n!!\; =\; (-1)^\backslash frac\{n-1\}\{2\}\; \backslash times\; n\backslash ,.$$

Disregarding the above definition of {{math|n!!}} for even values of {{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{{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}}{{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}}

$$\backslash begin\{align\}$$

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}

where $\backslash Gamma(z)$ 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, $n!!\; \backslash sim\; \backslash sqrt\{2\; n^\{n+1\}\; e^\{-n\}\}\backslash ,.$

The generalized formula $\backslash sqrt\{\backslash frac\{2\}\{\backslash pi\}\}\; 2^\backslash frac\{z\}\{2\}\; \backslash Gamma\backslash left(\backslash tfrac\; z2+1\backslash right)$ does not agree with the previous product formula for {{math|z!!}} for non-negative even integer values of {{mvar|z}}. Instead, this generalized formula implies the following alternative:

$$(2k)!!\; =\; \backslash sqrt\{\backslash frac\{2\}\{\backslash pi\}\}\; 2^k\; \backslash Gamma\backslash left(k+1\backslash right)\; =\; \backslash sqrt\{\; \backslash frac\{2\}\{\backslash pi\}\; \}\; \backslash prod\_\{i=1\}^k\; (2i)\; \backslash ,,$$

with the value for 0!! in this case being

{{startplainlist|indent=1}}

• $0!!\; =\; \backslash sqrt\{\; \backslash frac\{2\}\{\backslash pi\}\; \}\; \backslash approx\; 0.797\backslash ,884\backslash ,5608\backslash dots$ {{OEIS|id=A076668}}.

{{endplainlist}}

Using this generalized formula as the definition, the volume of an {{mvar|n}}-dimensional hypersphere of radius {{mvar|R}} can be expressed as{{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}}

$$V\_n=\backslash frac\{2\; \backslash left(2\backslash pi\backslash right)^\backslash frac\{n-1\}\{2\}\}\{n!!\}\; R^n\backslash ,.$$

For integer values of {{mvar|n}},

$$\backslash int\_\{0\}^\backslash frac\{\backslash pi\}\{2\}\backslash sin^n\; x\backslash ,dx=\backslash int\_\{0\}^\backslash frac\{\backslash pi\}\{2\}\backslash cos^n\; x\backslash ,dx=\backslash frac\{(n-1)!!\}\{n!!\}\backslash times$$

\begin{cases}1 & \text{if } n \text{ is odd} \\ \frac{\pi}{2} & \text{if } n \text{ is even.}\end{cases}

Using instead the extension of the double factorial of odd numbers to complex numbers, the formula is

$$\backslash int\_\{0\}^\backslash frac\{\backslash pi\}\{2\}\backslash sin^n\; x\backslash ,dx=\backslash int\_\{0\}^\backslash frac\{\backslash pi\}\{2\}\backslash cos^n\; x\backslash ,dx=\backslash frac\{(n-1)!!\}\{n!!\}\; \backslash sqrt\{\backslash frac\{\backslash pi\}\{2\}\}\backslash ,.$$

Double factorials can also be used to evaluate integrals of more complicated trigonometric polynomials.{{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}}

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

$$(2n-1)!!\; =\; 2^n\; \backslash cdot\; \backslash frac\{\backslash Gamma\backslash left(\backslash frac\{1\}\{2\}\; +\; n\backslash right)\}\; \{\backslash sqrt\{\backslash pi\}\}\; =\; (-2)^n\; \backslash cdot\; \backslash frac\{\backslash sqrt\{\backslash pi\}\}\; \{\; \backslash Gamma\backslash left(\backslash frac\{1\}\{2\}\; -\; n\backslash right)\}\backslash ,.$$

Some additional identities involving double factorials of odd numbers are:

$$\backslash begin\{align\}$$

(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}

An approximation for the ratio of the double factorial of two consecutive integers is

$$\backslash frac\{(2n)!!\}\{(2n-1)!!\}\; \backslash approx\; \backslash sqrt\{\backslash pi\; n\}.$$

This approximation gets more accurate as {{mvar|n}} increases, which can be seen as a result of the Wallis Integral.

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 {{mvar|α}}-factorial functions, extends the notion of the double factorial function for positive integers $\backslash alpha$:

$$$$

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}

Alternatively, the multifactorial {{math|z!_(α)}} 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

&= \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}

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!_(α)}} is not defined for negative multiples of {{mvar|α}}. However, it is defined and satisfies {{math|1=(z+α)!_(α) = (z+α)·z!_(α)}} for all other complex numbers {{mvar|z}}. This definition is consistent with the earlier definition only for those integers {{mvar|z}} satisfying {{math|z ≡ 1 mod α}}.

In addition to extending {{math|z!_(α)}} to most complex numbers {{mvar|z}}, this definition has the feature of working for all positive real values of {{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 alternative extension of the double factorial.

A class of generalized Stirling numbers of the first kind is defined for {{math|α > 0}} by the following triangular recurrence relation:

$$\backslash left[\backslash begin\{matrix\}\; n\; \backslash \backslash \; k\; \backslash end\{matrix\}\; \backslash right]\_\{\backslash alpha\}$$

= (\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}\,.

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)!_(α)}}, as

$$$$

\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}

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 ≡ n₀ mod α}} for {{math|n₀ ∈ {0, 1, 2, ..., α − 1}}}.

The generalized {{mvar|α}}-factorial polynomials, {{math|σ{{su|b=n|p=(α)}}(x)}} where {{math|σ{{su|b=n|p=(1)}}(x) ≡ σ_n(x)}}, which generalize the Stirling convolution polynomials from the single factorial case to the multifactorial cases, are defined by

$$\backslash sigma\_n^\{(\backslash alpha)\}(x)\; :=\; \backslash left[\backslash begin\{matrix\}\; x\; \backslash \backslash \; x-n\; \backslash end\{matrix\}\; \backslash right]\_\{(\backslash alpha)\}\; \backslash frac\{(x-n-1)!\}\{x!\}$$

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

$$\backslash sum\_\{n\; \backslash geq\; 0\}\; x\; \backslash cdot\; \backslash sigma\_n^\{(\backslash alpha)\}(x)\; z^n\; =\; e^\{(1-\backslash alpha)z\}\; \backslash left(\backslash frac\{\backslash alpha\; z\; e^\{\backslash alpha\; z\}\}\{e^\{\backslash alpha\; z\}-1\}\backslash right)^x\backslash ,.$$

Other combinatorial properties and expansions of these generalized {{mvar|α}}-factorial triangles and polynomial sequences are considered in {{harvtxt|Schmidt|2010}}.{{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}}

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)!_(α)}}, in terms of the Pochhammer symbol and the generalized, rational-valued binomial coefficients as

$$$$

\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}

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

$$$$

\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}

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}}.

$$(2n-1)!!\; =\; \backslash sum\_\{k=0\}^\{n-1\}\; \backslash binom\{n\}\{k+1\}\; (2k-1)!!\; (2n-2k-3)!!.$$

Similar identities can be obtained via context-free grammars.{{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 }} Additional finite sum expansions of congruences for the {{mvar|α}}-factorial functions, {{math|(αn − d)!_(α)}}, modulo any prescribed integer {{math|h ≥ 2}} for any {{math|0 ≤ d < α}} are given by {{harvtxt|Schmidt|2018}}.{{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}}

{{reflist}}

Category:Integer sequences

Category:Enumerative combinatorics

Category:Factorial and binomial topics

fr:Analogues de la factorielle#Multifactorielles