cross-base compatibility/uniqueness(?) - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: cross-base compatibility/uniqueness(?) (/showthread.php?tid=188) cross-base compatibility/uniqueness(?) - Gottfried - 07/14/2008 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/tetrationforum/showthread.php?tid=186&pid=2319#pid2319 [4] Helms, Gottfried; (binomial-method approximative equal to diagonalization) http://math.eretrandre.org/tetrationforum/showthread.php?tid=186&pid=2321#pid2321