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

[Gottfried]

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

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???

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?

Well, also I've the problem that I cannot go very deeply into this currently; what I've done the past days was to make some things straight (from my sketchy notes-pads) to put them out to you folks for consideration, because next days I'm absent and possibly take a complete rest for some weeks from mathematics at all (let's see).
I think the description of this formula in my article is pretty clear: what it gives..., what it is for... - only I did not yet discuss range of convergence (which I also use to extend by Euler-summation ... ) There is no other term needed and the base-parameter is just involved in the form of powers of log(base) , and the formula can be used for a certain range for log(base), whose bounds should be determined.

The "base-conversion" by this formula is surely not useful in general. This is, because it is only defined for integer iterates and it is much easier to compute with the integer iterates by the common function than by a powerseries... whose coefficients stem from a tetrated pascalmatrix. I could, for instance, think, that your's (and Jay's) may easily come out to be the reference-formula for this type of problem, why not.


Sorry I can't write more at the moment -

Regards -

Gottfried