Double factorial

In mathematics, the double factorial of a number n, denoted by n‼, is the product of all the positive integers up to n that have the same parity (odd or even) as n.[1] That is,

[5][6] The term semifactorial is also used by Knuth as a synonym of double factorial.

[7] In a 1902 paper, the physicist Arthur Schuster wrote:[8] 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,

for such products, and if a name be required for the product to call it the "alternate factorial" or the "double factorial".Meserve (1948)[9] 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.

[1][10] They occur in Student's t-distribution (1908), though Gosset did not use the double exclamation point notation.

Double factorials are motivated by the fact that they occur frequently in enumerative combinatorics and other settings.

For instance, n‼ for odd values of n counts Callan (2009) and 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 Rubey (2008) and Marsh & Martin (2011).

= ⁠1/3⁠; negative odd numbers with greater magnitude have fractional double factorials.

for even values of n, the double factorial for odd integers can be extended to most real and complex numbers z by noting that when z is a positive odd integer then[18][19]

The final expression is defined for all complex numbers except the negative even integers and satisfies (z + 2)!!

in this case being Using this generalized formula as the definition, the volume of an n-dimensional hypersphere of radius R can be expressed as[20]

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

[9][21] Double factorials of odd numbers are related to the gamma function by the identity:

Some additional identities involving double factorials of odd numbers are:[1]

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

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

Furthermore, when α = 1, this definition is mathematically equivalent to the Π(z) function, described above.

Also, when 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 α > 0 by the following triangular recurrence relation:

These generalized α-factorial coefficients then generate the distinct symbolic polynomial products defining the multiple factorial, or α-factorial functions, (x − 1)!

The distinct polynomial expansions in the previous equations actually define the α-factorial products for multiple distinct cases of the least residues x ≡ n0 mod α for n0 ∈ {0, 1, 2, ..., α − 1}.

The generalized α-factorial polynomials, σ(α)n(x) where σ(1)n(x) ≡ σn(x), which generalize the Stirling convolution polynomials from the single factorial case to the multifactorial cases, are defined by

These polynomials have a particularly nice closed-form ordinary generating function given by

Other combinatorial properties and expansions of these generalized α-factorial triangles and polynomial sequences are considered in Schmidt (2010).

Then we can expand the next single finite sums involving the multifactorial, or α-factorial functions, (αn − 1)!

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

The first two sums above are similar in form to a known non-round combinatorial identity for the double factorial function when α := 2 given by Callan (2009).

[23] Additional finite sum expansions of congruences for the α-factorial functions, (αn − d)!