The exponential factorial is a positive

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

''n'' raised to the power of ''n'' − 1, which in turn is raised to the power of ''n'' − 2, and so on 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_1 = 1,\quad  a_n = n^

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

262144 = 4^

Using the recurrence relation, the first exponential factorials are: :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 (mathematics), 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 ...

s or even hyperfactorials, in fact exhibiting tetrational growth. The number of digits in the exponential factorial of 6 is approximately 5 × 10^183 230. The sum of the reciprocals of the exponential factorials from 1 onwards is the following transcendental number: :

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

This sum is transcendental because it is a Liouville number. Like tetration, there is currently no accepted method of extension of the exponential factorial function to real 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 Γ, capital Greek alphabet, Greek letter gamma) is the most common extension of the factorial function to complex numbers. Derived by Daniel Bernoulli, the gamma function \Gamma(z) is defined ...

. But it is possible to expand it if it is defined in a strip width of 1. Similarly, there is disagreement about the appropriate value at 0; any value would be consistent with the recursive definition. A smooth extension to the reals would satisfy

f(0) = f'(1)

, which suggests a value strictly between 0 and 1.

Related functions, notation and conventions

Factorial In mathematics, the factorial of a non-negative denoted is the Product (mathematics), 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 ...

* Tetration

References

*Jonathan Sondow,
Exponential Factorial
From Mathworld, a Wolfram Web resource Factorial and binomial topics Integer sequences Large integers Exponentials {{Numtheory-stub