Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
cross-base compatibility/uniqueness(?)
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


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)

[4] Helms, Gottfried; (binomial-method approximative equal to diagonalization)
Gottfried Helms, Kassel

Possibly Related Threads...
Thread Author Replies Views Last Post
  A conjectured uniqueness criteria for analytic tetration Vladimir Reshetnikov 13 13,450 02/17/2017, 05:21 AM
Last Post: JmsNxn
  Uniqueness of half-iterate of exp(x) ? tommy1729 14 17,715 01/09/2017, 02:41 AM
Last Post: Gottfried
  Removing the branch points in the base: a uniqueness condition? fivexthethird 0 1,822 03/19/2016, 10:44 AM
Last Post: fivexthethird
  [2014] Uniqueness of periodic superfunction tommy1729 0 2,222 11/09/2014, 10:20 PM
Last Post: tommy1729
  Real-analytic tetration uniqueness criterion? mike3 25 24,831 06/15/2014, 10:17 PM
Last Post: tommy1729
  exp^[1/2](x) uniqueness from 2sinh ? tommy1729 1 2,597 06/03/2014, 09:58 PM
Last Post: tommy1729
  Uniqueness Criterion for Tetration jaydfox 9 12,464 05/01/2014, 10:21 PM
Last Post: tommy1729
  Uniqueness of Ansus' extended sum superfunction bo198214 4 7,489 10/25/2013, 11:27 PM
Last Post: tommy1729
  A question concerning uniqueness JmsNxn 3 6,244 10/06/2011, 04:32 AM
Last Post: sheldonison
  tetration bending uniqueness ? tommy1729 16 20,288 06/09/2011, 12:26 PM
Last Post: tommy1729

Users browsing this thread: 1 Guest(s)