I was considering remark of Dan Asimov, which Henryk cited in the thread "Bummer". Using my conversion-formula from T-tetration to U-tetration, which involves the fixpoints of the tetration-base b and application of the powerseries formula in my iteration-article, which I've announced here, I tend to conclude, that Asimov is right - because I get different powerseries, if I take different fixpoints, and where the related terms do not cancel pairwise.
Let b = t1^(1/t1) = t2^(1/t2) , u1=log(t1) and u2 = log(t2).
Then with the example b=sqrt(2), t1=2, u=u1=log(2), t2=4, u2=2*u and the conversion
Tb°h(x) = (Ut°h(x/t-1)+1)*t
I get for fixpoints t1 and t2 (and x=1 which is the assumption for tetration) the equation in question
(U2°h(-1/2)+1)*2 =?= (U4°h(-3/4)+1)*4
or rewritten
U2°h(-1/2) =?= U4°h(-3/4) + 1
The powerseries in u1 and u1^h resp u2 and u2^h seem so different, that one may possibly derive an argument from that, although I have not yet a convincing idea. Also possibly any general property of function-theory may be applicable then (for instance U2 is convergent and U4 is divergent, if I have it right).
See a better formatted short statement of this speculation here
[update] In a second view,and recalling the discussion and computations that we (and also myself) already had about the topic, the question is, why then is the difference so small? U_2 provides a convergent, U_4 a divergent series: the differences should be huge, if there are some, but: they are small. Hmmm...
Gottfried
Let b = t1^(1/t1) = t2^(1/t2) , u1=log(t1) and u2 = log(t2).
Then with the example b=sqrt(2), t1=2, u=u1=log(2), t2=4, u2=2*u and the conversion
Tb°h(x) = (Ut°h(x/t-1)+1)*t
I get for fixpoints t1 and t2 (and x=1 which is the assumption for tetration) the equation in question
(U2°h(-1/2)+1)*2 =?= (U4°h(-3/4)+1)*4
or rewritten
U2°h(-1/2) =?= U4°h(-3/4) + 1
The powerseries in u1 and u1^h resp u2 and u2^h seem so different, that one may possibly derive an argument from that, although I have not yet a convincing idea. Also possibly any general property of function-theory may be applicable then (for instance U2 is convergent and U4 is divergent, if I have it right).
See a better formatted short statement of this speculation here
[update] In a second view,and recalling the discussion and computations that we (and also myself) already had about the topic, the question is, why then is the difference so small? U_2 provides a convergent, U_4 a divergent series: the differences should be huge, if there are some, but: they are small. Hmmm...
Gottfried
Gottfried Helms, Kassel