Hi Sheldon -
I'm very sorry... I think I've expressed myself misleading.
The meaning of this formula,..., it does not relate two different bases. Only it gives a series for an iterate at a certain height and has the base-parameter b explicit (or: isolated) so we can work with it: extract it(possibly), build sums/series of powertowers of different bases, replace one base-parameter by another one.
So - although this is working with the base parameter, it is surely not what we are discussing with the focus on "change-of-base"-formula.
Sorry for mixing this up. (So also the question about the theta is inapplicable here.)
Btw, the capital P is simply the name for the Pascalmatrix, and P^^h with indexes means the entries of the first column of the h'th tetrated pascalmatrix, so the series gives just a(nother) definition of b^^h
Regards -
Gottfried
I'm very sorry... I think I've expressed myself misleading.
sheldonison Wrote:Gottfried Wrote:\( {b\^\^}^h = \sum_{k=0}^{\infty} \frac{{P\^\^}^h_{k,0}}{k!}\log(b)^k \)
....
The meaning of this formula,..., it does not relate two different bases. Only it gives a series for an iterate at a certain height and has the base-parameter b explicit (or: isolated) so we can work with it: extract it(possibly), build sums/series of powertowers of different bases, replace one base-parameter by another one.
So - although this is working with the base parameter, it is surely not what we are discussing with the focus on "change-of-base"-formula.
Sorry for mixing this up. (So also the question about the theta is inapplicable here.)
Btw, the capital P is simply the name for the Pascalmatrix, and P^^h with indexes means the entries of the first column of the h'th tetrated pascalmatrix, so the series gives just a(nother) definition of b^^h
Regards -
Gottfried
Gottfried Helms, Kassel
