A tiny base-dependent formula for tetration (change-of-base?)

[sheldonison]

Hi sheldonison (??? How can I adress you more personally?)

Sheldon

We need simply the first column of the tetrated pascalmatrix, scale it by reciprocal factorials and use it as coefficients for the powerseries:

\( {b\^\^}^h = \sum_{k=0}^{\infty} \frac{{P\^\^}^h_{k,0}}{k!}\log(b)^k \)
....

Gottfried,

Does your equation work for values of a,p > e^(1/e)?

?? No a,p in my formula ... What do you mean? Since you refer to e^(1/e) I assume you mean the base, but then you have "a,p" - how can this be a base???

typo, I meant b, p, the two bases in your equation.

Assuming it does, my next question would be does your equation need a \( b\^\^(h+\theta(h)) \) term?

Hmm. Again ??? "need your formula..." - for what?

....
The "base-conversion" by this formula is surely not useful in general. This is, because it is only defined for integer iterates ...
Regards -

Gottfried

The \( b\^\^(h+\theta(h)) \) term is only for real values, and only to meet the uniqueness criterion that the odd derivatives are all positive.