11/04/2014, 10:38 PM
a_n x^n = exp(exp(x)-1)
ln(a_n) + n ln(x) = exp(x)-1
ln(a_n) = Min[exp(x) - 1 - n ln(x)]
d/dx exp(x) - 1 - n ln(x) = exp(x) - n/x
exp(x) = n/x
x = W(n)
ln(a_n) = exp(W(n)) - 1 - n ln(W(n))
ln(a_n) = exp(W(n)) - 1 - n ln(n) - n W(n)
***
Need to compare this [ ln(a_n) ] to
ln(B_n) - lngamma(n+1)
for which many estimates exist.
However its not immediately clear which one to use.
Also a simplification of exp(W(n)) or exp(exp(W(n))) would be useful.
Does that exist ?
---
Many years ago I considered many forms of generalized bell numbers.
Together with " galathaea " and Han de Bruin this was partially discussed on sci.math.
One of those generalizations was
B_n(m) = D^n dexp^[m](x)
where n and m are positive integers , dexp is the decremented exponential [exp(x)-1] ,
D^n denotes the n th derivative and ^[m] is the m th iterate.
This might intrest you.
Its probably the most logical generalization from the viewpoint of tetration.
I notice that NOT MUCH IS EVER WRITTEN ABOUT TRIPLE EXPONENTIALS IN MATH !
---
More investigation is needed.
It seems hard to automate these ideas into a program or such.
On the other hand there is a clear direction to these ideas and inspiration is just around the corner.
regards
tommy1729
" Truth is what does not go away when you stop believing in it "
ln(a_n) + n ln(x) = exp(x)-1
ln(a_n) = Min[exp(x) - 1 - n ln(x)]
d/dx exp(x) - 1 - n ln(x) = exp(x) - n/x
exp(x) = n/x
x = W(n)
ln(a_n) = exp(W(n)) - 1 - n ln(W(n))
ln(a_n) = exp(W(n)) - 1 - n ln(n) - n W(n)
***
Need to compare this [ ln(a_n) ] to
ln(B_n) - lngamma(n+1)
for which many estimates exist.
However its not immediately clear which one to use.
Also a simplification of exp(W(n)) or exp(exp(W(n))) would be useful.
Does that exist ?
---
Many years ago I considered many forms of generalized bell numbers.
Together with " galathaea " and Han de Bruin this was partially discussed on sci.math.
One of those generalizations was
B_n(m) = D^n dexp^[m](x)
where n and m are positive integers , dexp is the decremented exponential [exp(x)-1] ,
D^n denotes the n th derivative and ^[m] is the m th iterate.
This might intrest you.
Its probably the most logical generalization from the viewpoint of tetration.
I notice that NOT MUCH IS EVER WRITTEN ABOUT TRIPLE EXPONENTIALS IN MATH !
---
More investigation is needed.
It seems hard to automate these ideas into a program or such.
On the other hand there is a clear direction to these ideas and inspiration is just around the corner.
regards
tommy1729
" Truth is what does not go away when you stop believing in it "
