Are those 2^3,2^3^4, 2^3^4^5... = 2^3^4^...n what Andrew calls E factorial as E(n) ? Does it have generalization to x? Obviously they can not start at 1. The smallest integer is 2, unlike ordinary factorial.
What if its turned around, so 2^1, 3^2^1, 4^3^2^1, 5^4^3^2^1.. n^..3^2^1. It is also a fast growing number. x^(x-1)^(x-2)..1, but much slower then the other.
This slower one has been called exponential factorial:
Exponential Factorial Wolfram MathWorld
It is given by recurence relation:
\( a_n=n^{a_{n-1}} \)
\( a_1=1 \)
Ramanujan's factorial would be bigger.
Do I understand right that by applying some transformation involving such factorials the summation of many divergent series can be brought to some sort of convergence-if their speed of growth is slower than these factorials?
Then these perhaps can be applied to power series of extremely slow functions directly, like e.g. 1/h(z).
Ivars
What if its turned around, so 2^1, 3^2^1, 4^3^2^1, 5^4^3^2^1.. n^..3^2^1. It is also a fast growing number. x^(x-1)^(x-2)..1, but much slower then the other.
This slower one has been called exponential factorial:
Exponential Factorial Wolfram MathWorld
It is given by recurence relation:
\( a_n=n^{a_{n-1}} \)
\( a_1=1 \)
Ramanujan's factorial would be bigger.
Do I understand right that by applying some transformation involving such factorials the summation of many divergent series can be brought to some sort of convergence-if their speed of growth is slower than these factorials?
Then these perhaps can be applied to power series of extremely slow functions directly, like e.g. 1/h(z).
Ivars