Английская Википедия:Double factorial

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Файл:Chord diagrams K6 matchings.svg

The fifteen different chord diagrams on six points, or equivalently the fifteen different perfect matchings on a six-vertex complete graph. These are counted by the double factorial Шаблон:Math.

In mathematics, the double factorial of a number Шаблон:Mvar, denoted by Шаблон:Math, is the product of all the positive integers up to Шаблон:Mvar that have the same parity (odd or even) as Шаблон:Mvar. 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 Шаблон:Mvar, the double factorial 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 Шаблон:Mvar 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, Шаблон:Math. The zero double factorial Шаблон:Math as an empty product.

The sequence of double factorials for even Шаблон:Mvar = Шаблон:Math starts as Шаблон:Block indent The sequence of double factorials for odd Шаблон:Mvar = Шаблон:Math starts as Шаблон:Block indent

The term odd factorial is sometimes used for the double factorial of an odd number.

The term semifactorial is also used by Knuth as a synonym of double factorial.

History and usage

In a 1902 paper, the physicist Arthur Schuster wrote: Шаблон:Quote

Шаблон:Harvtxt 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 t-distribution|Student's Шаблон:Mvar-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 Шаблон:Math, and it is much smaller than the iterated factorial Шаблон:Math.

The factorial of a positive Шаблон:Mvar may be written as the product of two double factorials: <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 Шаблон:Math.)

For an even non-negative integer Шаблон:Math with Шаблон:Math, the double factorial may be expressed as <math display="block">(2k)!! = 2^k k!\,.</math>

For odd Шаблон:Math with Шаблон:Math, 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 Шаблон:Math with Шаблон:Math, the double factorial may be expressed in terms of [[Permutation#Permutations without repetitions|Шаблон:Math-permutations of Шаблон:Math]] or a 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

Файл:Unordered binary trees with 4 leaves.svg

The fifteen different rooted binary trees (with unordered children) on a set of four labeled leaves, illustrating Шаблон:Math (see article text).

Double factorials are motivated by the fact that they occur frequently in enumerative combinatorics and other settings. For instance, Шаблон:Math for odd values of Шаблон:Mvar counts

• Perfect matchings of the complete graph Шаблон:Math for odd Шаблон:Mvar. In such a graph, any single vertex v has Шаблон:Mvar 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 Шаблон:Math items (permutations in which each cycle is a pair) or chord diagrams (sets of chords of a set of Шаблон:Math 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 Шаблон:Math in which each pair of equal numbers is separated only by larger numbers, where Шаблон:Math. The two copies of Шаблон:Mvar must be adjacent; removing them from the permutation leaves a permutation in which the maximum element is Шаблон:Math, with Шаблон:Mvar positions into which the adjacent pair of Шаблон:Mvar 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 Шаблон:Math positions of the permutation, so again the number of permutations may be counted by the double permutations.
• Heap-ordered trees, trees with Шаблон:Math nodes labeled Шаблон:Math, 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 Шаблон:Math labeled leaves. Each such tree may be formed from a tree with one fewer leaf, by subdividing one of the Шаблон:Mvar tree edges and making the new vertex be the parent of a new leaf.
• Rooted binary trees with Шаблон:Math 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.

Шаблон:Harvtxt and Шаблон:Harvtxt 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 Шаблон:Harvtxt and Шаблон:Harvtxt.

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 <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 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)!! = 1, (−3)!! = −1, and (−5)!! = Шаблон:Sfrac; negative odd numbers with greater magnitude have fractional double factorials. In particular, when Шаблон:Mvar 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 Шаблон:Math for even values of Шаблон:Mvar, the double factorial for odd integers can be extended to most real and complex numbers Шаблон:Mvar by noting that when Шаблон:Mvar is a positive odd integer then

<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 Шаблон:Math 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 Шаблон:Math for non-negative even integer values of Шаблон:Mvar. 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 Шаблон:Startplainlist

• <math>0!! = \sqrt{ \frac{2}{\pi} } \approx 0.797\,884\,5608\dots</math> Шаблон:OEIS.

Using this generalized formula as the definition, the volume of an Шаблон:Mvar-dimensional hypersphere of radius Шаблон:Mvar can be expressed as

<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 Шаблон:Mvar, <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.

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:

<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 Шаблон:Mvar 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 Шаблон:Mvar-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 Шаблон:Math can be extended to most real and complex numbers Шаблон:Math by noting that when Шаблон:Math is one more than a positive multiple of the positive integer Шаблон:Math 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 Шаблон:Math is not defined for negative integers, and Шаблон:Math is not defined for negative even integers, Шаблон:Math is not defined for negative multiples of Шаблон:Math. However, it is defined for and satisfies Шаблон:Math for all other complex numbers Шаблон:Math. This definition is consistent with the earlier definition only for those integers Шаблон:Math satisfying Шаблон:Math.

In addition to extending Шаблон:Math to most complex numbers Шаблон:Math, this definition has the feature of working for all positive real values of Шаблон:Math. Furthermore, when Шаблон:Math, this definition is mathematically equivalent to the Шаблон:Math function, described above. Also, when Шаблон:Math, 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 Шаблон:Math 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 Шаблон:Mvar-factorial coefficients then generate the distinct symbolic polynomial products defining the multiple factorial, or Шаблон:Mvar-factorial functions, Шаблон:Math, 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 Шаблон:Mvar-factorial products for multiple distinct cases of the least residues Шаблон:Math for Шаблон:Math.

The generalized Шаблон:Mvar-factorial polynomials, Шаблон:Math where Шаблон:Math, which generalize the 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 Шаблон:Math. 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 Шаблон:Mvar-factorial triangles and polynomial sequences are considered in Шаблон:Harvtxt.

Exact finite sums involving multiple factorial functions

Suppose that Шаблон:Math and Шаблон:Math are integer-valued. Then we can expand the next single finite sums involving the multifactorial, or Шаблон:Mvar-factorial functions, Шаблон:Math, 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 Шаблон:Math given by Шаблон:Harvtxt.

<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. Additional finite sum expansions of congruences for the Шаблон:Mvar-factorial functions, Шаблон:Math, modulo any prescribed integer Шаблон:Math for any Шаблон:Math are given by Шаблон:Harvtxt.