The exponential factorial is a positive integer ''n'' raised to the power of ''n'' − 1, which in turn is raised to the power of ''n'' − 2, and so on and so forth in a right-grouping manner. That is, :

n^

The exponential factorial can also be defined with the

recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

a_0 = 1,\quad  a_n = n^

The first few exponential factorials are 1, 1, 2, 9, 262144, etc. . For example, 262144 is an exponential factorial since :

262144 = 4^

Using the recurrence relation, the first exponential factorials are: :1 :1¹ = 1 :2¹ = 2 :3² = 9 :4⁹ = 262144 :5²⁶²¹⁴⁴ = 6206069878...8212890625 (183231 digits) The exponential factorials grow much more quickly than regular

factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \ ...

s or even

hyperfactorial In mathematics, and more specifically number theory, the hyperfactorial of a positive integer n is the product of the numbers of the form x^x from 1^1 to n^n. Definition The hyperfactorial of a positive integer n is the product of the numbers ...

s. The number of digits in the exponential factorial of 6 is approximately 5. The sum of the reciprocals of the exponential factorials from 1 onwards is the following

transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes ...

: :

\frac+\frac+\frac+\frac+\frac+\frac+\ldots=1.611114925808376736\underbrace_272243682859\ldots

This sum is transcendental because it is a

Liouville number In number theory, a Liouville number is a real number ''x'' with the property that, for every positive integer ''n'', there exists a pair of integers (''p, q'') with ''q'' > 1 such that :0 1 + \log_2(d) ~) no pair of integers ~(\,p,\,q\,)~ exists ...

. Like

tetration In mathematics, tetration (or hyper-4) is an operation based on iterated, or repeated, exponentiation. There is no standard notation for tetration, though \uparrow \uparrow and the left-exponent ''xb'' are common. Under the definition as rep ...

, there is currently no accepted method of extension of the exponential factorial function to

real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...

and

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

values of its argument, unlike the

function, for which such an extension is provided by the

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...

. But it is possible to expand it if it is defined in a strip width of 1.

Related functions, notation and conventions

References

*Jonathan Sondow,
Exponential Factorial
From

Mathworld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Di ...

, a Wolfram Web resource Factorial and binomial topics Integer sequences Large integers Exponentials {{Numtheory-stub