Re: Why is the tetration so much harder to extend than the factorial?

On Mar 4, 8:06 am, Gottfried Helms <he...@xxxxxxxxxxxxx> wrote:
Am 04.03.2009 13:48 schrieb G. A. Edgar:> In article
<dcdc9262-dc9f-4659-af0c-6c2c27aa3...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
galathaea <galath...@xxxxxxxxx> wrote:

gottfried helms has posted plenty of examples
where the method gives convergent series

An I wrong in my belief that he has never provided a proof of
convergence?

No, you're right. I was not concerned with proving so far.

To me it seemed, that it is even needed first to find a
reliable and (maximally) generalizable way to access the
problem at all, before invest in efforts to set out proofs.
(if I'd be capable of that at all...).

The matrix-based method for to deal with the formal powerseries
is good for many cases as shown by Comtet and later Woon, for two
instances, seems to be good even for series of powertowers.
But the reason for its failure in fractionally iterating with
bases b>e^(1/e) (using their complex fixpoints) is not yet
completely clear to me. So before I see a perspective there
(or a founded and explicite dismissal of the method for these
cases) I don't think I'll be sitting much to think about the
formal whereabouts and proofs...

Moreover, for instance in one aspect (fractional iteration of
exp(x)-1) divergence was already proved. How should I try to
prove convergence? What I try to find for these cases is a
method of summation for such strongly divergent series
instead of attempting to prove convergence. If I have a promising
one, valid for a reasonable range of parameters, I think I'll
try to deduce a proof (but likely I'll fail... :-))

From all discussion that I'm able to follow the impression
is, that we still are in a highly explorative situation,
so I think it is much helpful to collect facts, even if
isolated, from which -hopefully- a/"the" reliable, reasonably
general, framework can be derived (and be proved)

Currently I'm focused to the idea of iteration-series.
Maybe this gives such a framework for the whereabouts of
functional iteration and fractional tetration and can include
the unfriendly bases too (instead of getting lost in improper
complex powerseries) At least this had a certain inherent
logic...


What's "iteration series", anyway?
.

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