Double factorial

In mathematics, the double factorial or semifactorial of a number n, denoted by n,[1] is the product of all the integers from 1 up to n that have the same parity (odd or even) as n.[2] That is,

For example, 9‼ = 9 × 7 × 5 × 3 × 1 = 945. The zero double factorial 0‼ = 1 as an empty product.[3][4]

The sequence of double factorials for even n = 0, 2, 4, 6, 8,... starts as

The sequence of double factorials for odd n = 1, 3, 5, 7, 9,... starts as

The term odd factorial is sometimes used for the double factorial of an odd number.[5][6]

Meserve (1948)[7] (possibly the earliest publication to use double factorial notation)[8] 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.[2][9] 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 n!, and it is much smaller than the iterated factorial (n!)!.

The factorial of a non-zero n may be written as the product of two double factorials:[3]

where the denominator cancels the unwanted factors in the numerator. (The last form also applies when n = 0.)

For an even non-negative integer n = 2k with k ≥ 0, the double factorial may be expressed as

For odd n = 2k − 1 with k ≥ 1, combining the two above displays yields

For an odd positive integer n = 2k − 1 with k ≥ 1, the double factorial may be expressed in terms of k-permutations of 2k as[2][8]

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).[14][15]

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

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

Using this inverted recurrence, (−1)‼ = 1, (−3)‼ = −1, and (−5)‼ = 1/3; negative odd numbers with greater magnitude have fractional double factorials.[2] In particular, this gives, when n is an odd number,

Disregarding the above definition of n 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[16][17]

From this one can derive an alternative definition of z for non-negative even integer values of z:

The expression found for z is defined for all complex numbers except the negative even integers. Using it as the definition, the volume of an n-dimensional hypersphere of radius R can be expressed as[18]

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

Double factorials can also be used to evaluate integrals of more complicated trigonometric polynomials.[7][19]

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

Some additional identities involving double factorials of odd numbers are:[2]

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

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 α-factorial functions, extends the notion of the double factorial function for α ∈ ℤ+:

Alternatively, the multifactorial n!(α) can be extended to most real and complex numbers n by noting that when n is one more than a positive multiple of α then

This last expression is defined much more broadly than the original. In the same way that n! is not defined for negative integers, and n is not defined for negative even integers, n!(α) is not defined for negative multiples of α. However, it is defined for all other complex numbers. This definition is consistent with the earlier definition only for those integers n satisfying n ≡ 1 mod α.

In addition to extending n!(α) to most complex numbers n, this definition has the feature of working for all positive real values of α. Furthermore, when α = 1, this definition is mathematically equivalent to the Π(n) 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)!(α), as

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

for 0 ≤ nx. 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).[20]

Suppose that n ≥ 1 and α ≥ 2 are integer-valued. 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

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

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).

Additional finite sum expansions of congruences for the α-factorial functions, (αnd)!(α), modulo any prescribed integer h ≥ 2 for any 0 ≤ d < α are given by Schmidt (2017).[21]