10/02/2009, 02:55 AM
Hi.
I heard of a function called "exponential factorial" that works like this:
\( EF(1) = 1 \)
\( EF(2) = 2^1 = 2 \)
\( EF(3) = 3^{2^1} = 9 \)
\( EF(4) = 4^{3^{2^1}} = 262144 \)
\( EF(5) = 5^{4^{3^{2^1}}} = 5^{262144} \)
...
\( EF(n) = n^{EF(n-1)} \)
As one can see it is similar to tetration in that it involves a power tower, but it is not defined by iteration but by a different type of recurrence, similar to the factorial. Could there be a way to derive a smooth/analytic extension for this like there is with the factorial and gamma function and like how extensions have been proposed for tetration?
I heard of a function called "exponential factorial" that works like this:
\( EF(1) = 1 \)
\( EF(2) = 2^1 = 2 \)
\( EF(3) = 3^{2^1} = 9 \)
\( EF(4) = 4^{3^{2^1}} = 262144 \)
\( EF(5) = 5^{4^{3^{2^1}}} = 5^{262144} \)
...
\( EF(n) = n^{EF(n-1)} \)
As one can see it is similar to tetration in that it involves a power tower, but it is not defined by iteration but by a different type of recurrence, similar to the factorial. Could there be a way to derive a smooth/analytic extension for this like there is with the factorial and gamma function and like how extensions have been proposed for tetration?
