Half-iterate exp(z)-1: hypothese on growth of coefficients

[MphLee]

Well, I just put a vague idea... If we separate the powerseries of the exp() in two partial series, then the partial series are as well interesting functions (sinh() and cosh() )  and if in the two partial series we change each second coefficient's sign we have sin() and cos() instead, which relate to sinh() and cosh() by using imaginary instead real arguments. As said: just a vague idea, perhaps there might something similar interesting come out ... if researched effectively...   

Wow, this sounds suggestive. Like tetrational trig functions or something like that, since we can use \(f\) to recover a tetration.

But the central topic is: 1) do we know a functional bound for the coefficients depending on the index \( k\) at all? (I.N. Baker said in his 1958 article: "we have no apriori knowledge about their growthrate" and I've never seen some estimate of this) 2) can we use a known summation method, for instance Borel-summation, such that we have also a formally proved arbitrarily-approximatable procedure, or that we can build a new one based on our knowledge of that specific growth-rate. - - -  (Btw.: the separation into four partial series is not really important here: the limiting is of course valid for the unsegmented series as well)  A big plan, likely too big for me, but my heuristic is simply too nice...

Maybe it is important because it allows one to express the overall chaos as a sum/superposition of waves of with different phases...? It is the ignorant part of me talking here.

:-) Yes, sure. Has been too much for me to investigate seriously when I had also to do my job. And to accompany my family. And to be pappa for my child in home. :-)

The important things first... even if sometimes math can lure us into thinking she is our Beatrice... bring us to hell...

Nothing to problematize. I thought only about something like for a complete beginner, how such summations work - in principle, with examples. If sometimes someone liked that idea he/she might come back to this :-)

Ok, if more ppl are interested count me in, theoretically... in practice I'll be in vacation very soon so I'll be back in September... and then I don't know what will be my situation for what concerns weekends and days off from work..

And while I'm scanning my old hobby-treatizes I think there are some even better workouts, however I never completed a stand-alone text on this subject.
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To make the matter now a bit less prominent: having strong loss of energy since several months now I'm only skimming through my early elaborations and word-docs and if I find something worth to be looked at, which I did not present earlier here in the forum (but which is related to our matter), I think I'll add it "to the pipeline" for someone who might be interested by chance (or has a knack for historical matters)... Hmm, maybe a subforum for such thoughts might even be more appropriate...

Given my lack of time for serious research I also started during the last year or so skimming trhu my old stuff... trying to save things and also making archives of interesting matierial. For example I have a complete archive of Romerio's, Rubtsov's, Trappmann's and JmsNxn's papers. Also was able to save most of the literature of this forum about hos and iteration. I was starting these weeks trying to complete my archive also with all your works(eg. on matrices) and most of my forum contributions.

Some questions arise... I'm saving all this stuff... will I ever be able to read/use it? xD But I like the idea that I'm saving knowledge.
Another question is: was my contribute and my body of notes worth of something? hahah damn... I'm not old (not young anymore) yet I feel the pressure of time.

All of this to say that... idk what do you have in mind.. I love historical matters because I think that the historical perspective is often the best order to impose on complex matters that makes things comprehensible.
So just open a new thread about that subforum idea, let's talk about it.