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)
\( \hspace{24} T_b^{\circ h}(x) = (U_t^{\circ h} (\frac{x}{t}-1) + 1)*t \)
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 \( t_2=2 \) and \( t_4=4 \) such that \( t_2^ {1/t_2} = t_4^ {1/t_4} = b \)
--------------------------
We expect then, for general height h,
\( \hspace{24} \begi{eqnarray}
T_b^{\circ h}(x)& =& (U_2^{\circ h}(\frac{x}{2}-1)+1)*2 \\
& =& (U_4^{\circ h}(\frac{x}{4}-1)+1)*4
\end{eqnarray} \)
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
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)
\( \hspace{24} T_b^{\circ h}(x) = (U_t^{\circ h} (\frac{x}{t}-1) + 1)*t \)
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 \( t_2=2 \) and \( t_4=4 \) such that \( t_2^ {1/t_2} = t_4^ {1/t_4} = b \)
--------------------------
We expect then, for general height h,
\( \hspace{24} \begi{eqnarray}
T_b^{\circ h}(x)& =& (U_2^{\circ h}(\frac{x}{2}-1)+1)*2 \\
& =& (U_4^{\circ h}(\frac{x}{4}-1)+1)*4
\end{eqnarray} \)
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
