Mizugadro, pentation, Book

[Gottfried]

Hello Dmitrii -

If I find it I'd try this using Pari/GP to see myself, trying to use greater precision, whether my approach can be finetuned to arrive at your values at all.

I think, first we should compare our results for some fixed value of base,
for example, for natural tetration , or for b=sqrt(2).
Then we may consider the derivatives with respect to base.

Well, here I see no problem; as you write that you use for that the regular tetration (which employs power series centered around the lower fixpoint (which is real)) - and that is nicely expressible by the Carlemanmatrix-algebra. So here we have automatically identity because the methods match even formally.

That simple-to-formulate Carlemanmatrix-approach ( for the base 1<b<exp(1/e) using diagonalization to implement the regular tetration) can be simply extended for the limit-case base b=exp(1/e) using matrix-logarithm instead of the (well understood) diagonalization.
Only I did not know whether your approach to that specific base (let alone to the larger bases) could be reproduced by that matrix-log algebra . With that matrix-log-algebra the well known discussion concerning the iteration of f(x) = exp(x)-1 using its formal power series is properly reproduced and the -again well known- divergence of its powerseries at fractional height iterations is (hypothetically) overcome by the method of Noerlund-summation.
What I hoped/hope for was/is, that your method might be stronger (making the computation of actual values simpler than that divergent summation process), so my hope was, that I can adapt that quasi-standard procedures formally by some tweak to arrive at your results.
So I'll see, whether I can reproduce the values using your procedure of "primitive fit" and then how that can then possibly be expressed by building new elements on the conventional/known processes.

But that might now take some time, sigh...

Thanks first so far for your hints -

Gottfried

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update
By the time I've computed approximate estimates for the base b=exp(1/e) tetration (always starting at z0=1) with iteration heights h=0 to 0.5. I've got this table of results

Code:

------------------------------------
     h      tet_b(h)
------------------------------------
    0.00                      1
    0.05  1.0299867588584812146
    0.10  1.058880695285946978'0
    0.15  1.08674287936747424'48
    0.20  1.1136297821277916'69
    0.25  1.139593709988212'70
    0.30  1.16468319003852'55
    0.35  1.1889433125893'41
    0.40  1.212416036506'10
    0.45  1.23514046202'20    
    0.50  1.2571530750'54

where the quote-symbol ' indicates the achieved trusted precision using summation of series with 128 terms to arrive at the coefficients c_k and 16 coefficients c_k are computed to form the polynomial which estimates the true tet_b(h) for the iteration-height argument h using the leading 16 coefficients