I should also add a remark, which I've made already. This is the link to the Euler-paper E190 which seem to deal with a false series for logarithm, in the sense:

s_a = 1

s_a°(h+1) = a*s_a°h

for integer h - which means, this series mimics the logarithm of a^h - for integer h exclusively.

This series looks much like the series, which we get if we use the fixpoint-shift and evaluate iterates: exact results with integer heights, false results with algebraic heights. Perhaps its series-inverse is even a closer related series to our tetration-series (as given in the symbolic eigensystem-decomposition), and we are on a false track, when employing the fixpoint-shift.

Gottfried

s_a = 1

s_a°(h+1) = a*s_a°h

for integer h - which means, this series mimics the logarithm of a^h - for integer h exclusively.

This series looks much like the series, which we get if we use the fixpoint-shift and evaluate iterates: exact results with integer heights, false results with algebraic heights. Perhaps its series-inverse is even a closer related series to our tetration-series (as given in the symbolic eigensystem-decomposition), and we are on a false track, when employing the fixpoint-shift.

Gottfried

Gottfried Helms, Kassel