Factorial

The factorial of 0 is 1, or in symbols, 0! = 1.

There are several motivations for this definition:

• For n = 0, the definition of n! as a product involves the product of no numbers at all, and so is an example of the broader convention that the product of no factors is equal to the multiplicative identity (see Empty product).
• There is exactly one permutation of zero objects (with nothing to permute, the only rearrangement is to do nothing).
• It makes many identities in combinatorics valid for all applicable sizes. The number of ways to choose 0 elements from the empty set is given by the binomial coefficient

${\displaystyle {\binom {0}{0}}={\frac {0!}{0!0!}}=1.}$

More generally, the number of ways to choose all n elements among a set of n is

${\displaystyle {\binom {n}{n}}={\frac {n!}{n!0!}}=1.}$

• It allows for the compact expression of many formulae, such as the exponential function, as a power series:

${\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}.}$

• It extends the recurrence relation to 0.

The gamma function interpolates the factorial function to non-integer values. The main clue is the recurrence relation generalized to a continuous domain.

Besides nonnegative integers, the factorial can also be defined for non-integer values, but this requires more advanced tools from mathematical analysis.

One function that fills in the values of the factorial (but with a shift of 1 in the argument), that is often used, is called the gamma function, denoted Γ(z). It is defined for all complex numbers z except for the non-positive integers, and given when the real part of z is positive by

${\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}e^{-t}\,dt.}$

Its relation to the factorial is that n! = Γ(n + 1) for every nonnegative integer n.

Euler's original formula for the gamma function was

${\displaystyle \Gamma (z)=\lim _{n\to \infty }{\frac {n^{z}n!}{\displaystyle \prod _{k=0}^{n}(z+k)}}.}$

Carl Friedrich Gauss used the notation Π(z) to denote the same function, but with argument shifted by 1, so that it agrees with the factorial for nonnegative integers. This pi function is defined by

${\displaystyle \Pi (z)=\int _{0}^{\infty }t^{z}e^{-t}\,dt.}$

The pi function and gamma function are related by the formula Π(z) = Γ(z + 1). Likewise, Π(n) = n! for any nonnegative integer n.

The factorial function, generalized to all real numbers except negative integers. For example, 0! = 1! = 1, (−1/2)! = √π, 1/2! = √π/2.

In addition to this, the pi function satisfies the same recurrence as factorials do, but at every complex value z where it is defined

${\displaystyle \Pi (z)=z\Pi (z-1)\,.}$

This is no longer a recurrence relation but a functional equation. In terms of the gamma function, it is

${\displaystyle \Gamma (n+1)=n\Gamma (n)\,.}$

The values of these functions at half-integer values is therefore determined by a single one of them:

${\displaystyle \Gamma \left({\frac {1}{2}}\right)=\left(-{\frac {1}{2}}\right)!=\Pi \left(-{\frac {1}{2}}\right)={\sqrt {\pi }}\,,}$

from which it follows that for n ∈ N,

${\displaystyle {\begin{aligned}&\Gamma \left({\frac {1}{2}}+n\right)=\left(-{\frac {1}{2}}+n\right)!=\Pi \left(-{\frac {1}{2}}+n\right)\\[5pt]={}&{\sqrt {\pi }}\prod _{k=1}^{n}{\frac {2k-1}{2}}={\frac {(2n)!}{4^{n}n!}}{\sqrt {\pi }}={\frac {(2n-1)!}{2^{2n-1}(n-1)!}}{\sqrt {\pi }}\,.\end{aligned}}}$

For example,

${\displaystyle \Gamma \left({\frac {9}{2}}\right)={\frac {7}{2}}!=\Pi \left({\frac {7}{2}}\right)={\frac {1}{2}}\cdot {\frac {3}{2}}\cdot {\frac {5}{2}}\cdot {\frac {7}{2}}{\sqrt {\pi }}={\frac {8!}{4^{4}4!}}{\sqrt {\pi }}={\frac {7!}{2^{7}3!}}{\sqrt {\pi }}={\frac {105}{16}}{\sqrt {\pi }}\approx 11.631\,728\ldots }$

It also follows that for n ∈ N,

${\displaystyle \Gamma \left({\frac {1}{2}}-n\right)=\left(-{\frac {1}{2}}-n\right)!=\Pi \left(-{\frac {1}{2}}-n\right)={\sqrt {\pi }}\prod _{k=1}^{n}{\frac {2}{1-2k}}={\frac {\left(-4\right)^{n}n!}{(2n)!}}{\sqrt {\pi }}\,.}$

For example,

${\displaystyle \Gamma \left(-{\frac {5}{2}}\right)=\left(-{\frac {7}{2}}\right)!=\Pi \left(-{\frac {7}{2}}\right)={\frac {2}{-1}}\cdot {\frac {2}{-3}}\cdot {\frac {2}{-5}}{\sqrt {\pi }}={\frac {\left(-4\right)^{3}3!}{6!}}{\sqrt {\pi }}=-{\frac {8}{15}}{\sqrt {\pi }}\approx -0.945\,308\ldots }$

The pi function is certainly not the only way to extend factorials to a function defined at almost all complex values, and not even the only one that is analytic wherever it is defined. Nonetheless it is usually considered the most natural way to extend the values of the factorials to a complex function. For instance, the Bohr–Mollerup theorem states that the gamma function is the only function that takes the value 1 at 1, satisfies the functional equation Γ(n + 1) = nΓ(n), is meromorphic on the complex numbers, and is log-convex on the positive real axis. A similar statement holds for the pi function as well, using the Π(n) = nΠ(n − 1) functional equation.

However, there exist complex functions that are probably simpler in the sense of analytic function theory and which interpolate the factorial values. For example, Hadamard's 'gamma' function (Hadamard 1894) which, unlike the gamma function, is an entire function.[22]

Euler also developed a convergent product approximation for the non-integer factorials, which can be seen to be equivalent to the formula for the gamma function above:

${\displaystyle {\begin{aligned}n!=\Pi (n)&=\prod _{k=1}^{\infty }\left({\frac {k+1}{k}}\right)^{n}\!\!{\frac {k}{n+k}}\\&=\left[\left({\frac {2}{1}}\right)^{n}{\frac {1}{n+1}}\right]\left[\left({\frac {3}{2}}\right)^{n}{\frac {2}{n+2}}\right]\left[\left({\frac {4}{3}}\right)^{n}{\frac {3}{n+3}}\right]\cdots \end{aligned}}}$

However, this formula does not provide a practical means of computing the pi function or the gamma function, as its rate of convergence is slow.

The volume of an n-dimensional hypersphere of radius R is

${\displaystyle V_{n}={\frac {\pi ^{n/2}}{\Gamma \left({\frac {n}{2}}+1\right)}}R^{n}\,.}$

Amplitude and phase of factorial of complex argument

Representation through the gamma function allows evaluation of factorial of complex argument. Equilines of amplitude and phase of factorial are shown in figure. Let

${\displaystyle f=\rho e^{i\varphi }=(x+iy)!=\Gamma (x+iy+1)\,.}$

Several levels of constant modulus (amplitude) ρ and constant phase φ are shown. The grid covers the range −3 ≤ x ≤ 3, −2 ≤ y ≤ 2, with unit steps. The scratched line shows the level φ = ±π.

Thin lines show intermediate levels of constant modulus and constant phase. At the poles at every negative integer, phase and amplitude are not defined. Equilines are dense in vicinity of singularities along negative integer values of the argument.

For |z| < 1, the Taylor expansions can be used:

${\displaystyle z!=\sum _{n=0}^{\infty }g_{n}z^{n}\,.}$

The first coefficients of this expansion are

where γ is the Euler–Mascheroni constant and ζ is the Riemann zeta function. Computer algebra systems such as SageMath can generate many terms of this expansion.

For the large values of the argument, the factorial can be approximated through the integral of the digamma function, using the continued fraction representation. This approach is due to T. J. Stieltjes (1894). Writing z! = e^P(z) where P(z) is

${\displaystyle P(z)=p(z)+{\frac {\ln 2\pi }{2}}-z+\left(z+{\frac {1}{2}}\right)\ln(z)\,,}$

Stieltjes gave a continued fraction for p(z):

${\displaystyle p(z)={\cfrac {a_{0}}{z+{\cfrac {a_{1}}{z+{\cfrac {a_{2}}{z+{\cfrac {a_{3}}{z+\ddots }}}}}}}}}$

The first few coefficients a_n are[23]

n	an
0	1/12
1	1/30
2	53/210
3	195/371
4	22999/22737
5	29944523/19733142
6	109535241009/48264275462

There is a misconception that ln z! = P(z) or ln Γ(z + 1) = P(z) for any complex z ≠ 0. Indeed, the relation through the logarithm is valid only for a specific range of values of z in the vicinity of the real axis, where −π < Im(Γ(z + 1)) < π. The larger the real part of the argument, the smaller the imaginary part should be. However, the inverse relation, z! = e^P(z), is valid for the whole complex plane apart from z = 0. The convergence is poor in the vicinity of the negative part of the real axis; it is difficult to have good convergence of any approximation in the vicinity of the singularities. When |Im z| > 2 or Re z > 2, the six coefficients above are sufficient for the evaluation of the factorial with complex double precision. For higher precision more coefficients can be computed by a rational QD scheme (Rutishauser's QD algorithm).[24]

The relation n! = n × (n − 1)! allows one to compute the factorial for an integer given the factorial for a smaller integer. The relation can be inverted so that one can compute the factorial for an integer given the factorial for a larger integer:

${\displaystyle (n-1)!={\frac {n!}{n}}.}$

However, this recursion does not permit us to compute the factorial of a negative integer; use of the formula to compute (−1)! would require a division of a nonzero value by zero, and thus blocks us from computing a factorial value for every negative integer. Similarly, the gamma function is not defined for zero or negative integers, though it is defined for all other complex numbers.

The product of all the odd integers up to some odd positive integer n is called the double factorial of n, and denoted by n!!.[25] That is,

${\displaystyle (2k-1)!!=\prod _{i=1}^{k}(2i-1)={\frac {(2k)!}{2^{k}k!}}={\frac {_{2k}P_{k}}{2^{k}}}={\frac {\left(2k\right)^{\underline {k}}}{2^{k}}}\,.}$

For example, 9!! = 1 × 3 × 5 × 7 × 9 = 945.

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

1, 3, 15, 105, 945, 10395, 135135,... (sequence A001147 in the OEIS)

Double factorial notation may be used to simplify the expression of certain trigonometric integrals,[26] to provide an expression for the values of the gamma function at half-integer arguments and the volume of hyperspheres,[27] and to solve many counting problems in combinatorics including counting binary trees with labeled leaves and perfect matchings in complete graphs.[25][28]

A common related notation is to use multiple exclamation points to denote a multifactorial, the product of integers in steps of two (n!!), three (n!!!), or more (see generalizations of the double factorial). The double factorial is the most commonly used variant, but one can similarly define the triple factorial (n!!!) and so on. One can define the k-tuple factorial, denoted by n!^(k), recursively for positive integers as

${\displaystyle n!^{(k)}={\begin{cases}n&{\text{if }}0<n\leq k,\\n\left({(n-k)!}^{(k)}\right)&{\text{if }}n>k.\end{cases}}}$

In addition, similarly to 0! = 1!/1 = 1, one can define:

${\displaystyle {n!}^{(k)}=1\quad {\text{if}}\ {-k}<n\leq 0.}$

For sufficiently large n ≥ 1, the ordinary single factorial function is expanded through the multifactorial functions as follows:

${\displaystyle {\begin{aligned}n!&=n!!\cdot (n-1)!!\,,&n&\geq 1\\[5px]&=n!!!\cdot (n-1)!!!\cdot (n-2)!!!\,,&n&\geq 2\\[5px]&=\prod _{i=0}^{k-1}(n-i)!^{(k)},\quad {\text{ for }}k\in \mathbb {Z} ^{+}\,,&n&\geq k-1.\end{aligned}}}$

In the same way that n! is not defined for negative integers, and n!! is not defined for negative even integers, n!^(k) is not defined for negative integers divisible by k.

The primorial of a natural number n (sequence A002110 in the OEIS), denoted n#, is similar to the factorial, but with the product taken only over the prime numbers less than or equal to n. That is,

${\displaystyle n\#=\prod _{p\leq n}p,}$

where p ranges over the prime numbers less than or equal to n. For example, the primorial of 11 is

${\displaystyle 11\#=11\times 7\times 5\times 3\times 2=2310.}$

Neil Sloane and Simon Plouffe defined a superfactorial in The Encyclopedia of Integer Sequences (Academic Press, 1995) to be the product of the first n factorials. So the superfactorial of 4 is

${\displaystyle \operatorname {sf} (4)=1!\times 2!\times 3!\times 4!=288\,.}$

In general

${\displaystyle {\begin{aligned}\operatorname {sf} (n)=\prod _{k=1}^{n}k!&=\prod _{k=1}^{n}k^{n-k+1}\\&=1^{n}\cdot 2^{n-1}\cdot 3^{n-2}\cdots (n-1)^{2}\cdot n^{1}\,.\end{aligned}}}$

Equivalently, the superfactorial is given by the formula

${\displaystyle \operatorname {sf} (n)=\prod _{0\leq i<j\leq n}(j-i)}$

which is the determinant of a Vandermonde matrix.

The superfactorials can be extended to all complex numbers with the Barnes G-function, such that ${\displaystyle G(n)=\operatorname {sf} (n)}$ for all positive integers n. The sequence of superfactorials starts (from n = 0) as

1, 1, 2, 12, 288, 34560, 24883200, 125411328000,... (sequence A000178 in the OEIS)

By this definition, we can define the k-superfactorial of n (denoted sf_k(n)) as:

${\displaystyle \operatorname {sf} _{k}(n)={\begin{cases}n&{\text{if }}k=0\\\prod _{r=1}^{n}\operatorname {sf} _{k-1}(r)&{\text{if }}k\geq 1\end{cases}}}$

The 2-superfactorials of n are

1, 1, 2, 24, 6912, 238878720, 5944066965504000, 745453331864786829312000000,... (sequence A055462 in the OEIS)

The 0-superfactorial of n is n.

In his 1995 book Keys to Infinity, Clifford Pickover defined a different function n$ that he called the superfactorial. It is defined by

${\displaystyle n\$\equiv {\begin{matrix}\underbrace {n!^{{n!}^{{\cdot }^{{\cdot }^{{\cdot }^{n!}}}}}} \\n!{\mbox{ copies of }}n!\end{matrix}}.}$

This sequence of superfactorials starts

${\displaystyle {\begin{aligned}1\__SUB_LEVEL_SECTION_19__amp;=1\,,\\2\__SUB_LEVEL_SECTION_19__amp;=2^{2}=4\,,\\3\__SUB_LEVEL_SECTION_19__amp;=6^{6^{6^{6^{6^{6}}}}}\,.\end{aligned}}}$

(Here, as is usual for compound exponentiation, the grouping is understood to be from right to left: a^bc = a^(bc).)

This operation may also be expressed as the tetration

${\displaystyle n\$={}^{n!}(n!)\,,}$

or using Knuth's up-arrow notation as

${\displaystyle n\$=(n!)\uparrow \uparrow (n!)=(n!)\uparrow \uparrow \uparrow 2\,.}$

Occasionally the hyperfactorial of n is considered. It is written as H(n) and defined by

${\displaystyle {\begin{aligned}H(n)&=\prod _{k=1}^{n}k^{k}\\&=1^{1}\cdot 2^{2}\cdot 3^{3}\cdots (n-1)^{n-1}\cdot n^{n}.\end{aligned}}}$

For n = 1, 2, 3, 4,... the values of H(n) are 1, 4, 108, 27648,... (sequence A002109 in the OEIS).

The asymptotic growth rate is

${\displaystyle H(n)\sim An^{(6n^{2}+6n+1)/12}e^{-n^{2}/4}}$

where A = 1.2824... is the Glaisher–Kinkelin constant.[29] H(14) ≈ 1.8474×10⁹⁹ is already almost equal to a googol, and H(15) ≈ 8.0896×10¹¹⁶ is almost of the same magnitude as the Shannon number, the theoretical number of possible chess games. Compared to the Pickover definition of the superfactorial, the hyperfactorial grows relatively slowly.

The hyperfactorial function can be generalized to complex numbers in a similar way as the factorial function. The resulting function is called the K-function.