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 "