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cross-base compatibility/uniqueness(?)
#1
In a discussion in sci.math I introduced the term "cross-base-compatibility" for tetration which is thought to implement another restriction on fractional iteration, which possibly makes it unique.
Here I cite myself with two articles in sci.math (a little bit edited), melting them here into one.


(...)

b) Interpolation: in a previous post I discussed a likely difference of methods, when different interpolation-approaches depending on the h-parameter are assumed.
You gave a polynomial interpolation approach, which I think is somehow natural.
But the coefficients at -for instance- x^1 with increasing h

(1,1,1,1,1,...)

or at x^2

(0,1/2,2/2,3/2,4/2,... )

can also be interpolated including -for instance- a sine-function of h; as well as the coefficients at higher powers of x.

I don't mean to play a game of obfuscation here: the reason for my being "not-completely-satisfied" is, that -using U-tetration as application of T-tetration with fixpoint-shift- the common tetration (T-tetration in my wording) seems to be dependent on the selection of the fixpoint, if fractional iterations are computed.

So - while the matrix-based (diagonalization) method using U-tetration for each fixed base only may be consistent, when the polynomial interpolation-approach is applied, then still the cross-base-relations might be "imperfect"...

Hmm - I must be vague this way, because I still don't have hard data at hand to see these differences (they are said to be small, may be smaller than my approximation-accuracy) and so seem currently to be too small to be able to experiment with this problem effectively.

At least there is one consolidation: in my previous post I mentioned the different interpolation-method using the binomial-expansion and values of the powertower-function themselves (as can be seen for instance in [1],[2] or [3]) Here I found different results in my first comparision (using insufficient approximation) - however, a new computation indicates now, that the results of this method and of the diagonalization may come out to be the same (as expected) [4]


(second letter)

(...)
Perhaps I should explain this a bit more.

The fixpoint-substitution, which relates T and U-tetration is, for a base b=t^(1/t)



for the integer case of h; for fractional this is then assumed.

We try, using the most simple case, base b=sqrt(2) = 2^(1/2) = 4^(1/4)

So let and such that

--------------------------

We expect then, for general height h,


so the U-tetration for base 2 and for base 4 must give "compatible" results for their fractional interpolations - this is what I meant with "cross-base-relations"

The series, which occur with these U-tetrates are all divergent, and I can assign values only to a certain accuracy - while a summation-method were needed, which allows arbitrary accuracy: to first quantify the difference according to the different fixpoint-shifts and then second to formulate a hypothese for a correction-term, which is worth to work on.



[1] Comtet, Louis; Advanced Combinatorics,

[2] Woon, S.C.; Analytic Continuation of Operators —
Operators acting complex s-times
Chap 9 (online available in arXiv-org)

[3] Robbins, Andrew; (forum-message binomial-method=Woon-method)
http://math.eretrandre.org/tetrationforu...19#pid2319

[4] Helms, Gottfried; (binomial-method approximative equal to diagonalization)
http://math.eretrandre.org/tetrationforu...21#pid2321
Gottfried Helms, Kassel
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