Theexponential factorial is a positiveintegernraised to the power ofn − 1, which in turn is raised to the power ofn − 2, and so on in a right-grouping manner. That is,
The exponential factorial can also be defined with therecurrence relation
The first few exponential factorials are1,2,9,262144, ... (OEIS: A049384 orOEIS: A132859). For example, 262144 is an exponential factorial since
Using the recurrence relation, the first exponential factorials are:
The exponential factorials grow much more quickly than regularfactorials or evenhyperfactorials, in fact exhibiting tetrational growth. The number of digits in the exponential factorial of 6 is approximately 5 × 10183 230.
The sum of thereciprocals of the exponential factorials from 1 onwards is the followingtranscendental number:
This sum is transcendental because it is aLiouville number.
Liketetration, there is currently no accepted method of extension of the exponential factorial function toreal andcomplex values of its argument, unlike thefactorial function, for which such an extension is provided by thegamma function. 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, which suggests a value strictly between 0 and 1.
 | Thisnumber theory-related article is astub. You can help Wikipedia byexpanding it. |