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Factorials grow faster than exponential functions, but much more slowly than double exponential functions. However, tetration and the Ackermann function grow faster. See Big O notation for a comparison of the rate of growth of various functions. The inverse of the double exponential function is the double logarithm log(log(x)).
Just as triangular numbers sum the numbers from to , and factorials take their product, the exponential factorial exponentiates. The exponential factorial is defined recursively as =, =. For example, the exponential factorial of 4 is = These numbers grow much more quickly than regular factorials. [95] Falling factorial
Comparison of Stirling's approximation with the factorial. In mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate results even for small values of .
An alternative notation for the rising factorial () is the less common () +. When () + is used to denote the rising factorial, the notation () is typically used for the ordinary falling factorial, to avoid confusion. [3]
The exponential factorials grow much more quickly than regular factorials or even hyperfactorials. 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:
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Just as the factorials can be continuously interpolated by the gamma function, the superfactorials can be continuously interpolated by the Barnes G-function. [ 2 ] According to an analogue of Wilson's theorem on the behavior of factorials modulo prime numbers, when p {\displaystyle p} is an odd prime number s f ( p − 1 ) ≡ ( p − 1 ) ! !
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