Sum formula and "value at -1"

Is it possible that the continuum sum formula of Ansus

\( \frac{\mathrm{tet}'(z + z_0)}{\mathrm{tet}'(z_0)} = \exp\left(\sum_{n=0}^{z-1} \mathrm{tet}(n + z_0)\right) \)


\( f(x) = K \int_{-1}^{x} (\log(b))^t \exp_b\left(\sum_{k=0}^{t-1} f(k)\right) dt \).

can be iterated with only the function in a small region about x = 0 or -1... i.e. is there some way to do the integral or differentiation without knowing the values at x = 0 or -1 explicitly? Suppose a method is found that can iterate the sum operator on a Taylor series, but there is a singularity too close to 0 (within radius 1 of 0) that it cannot converge at -1. Consider base 1/4. As a Taylor series at 0, I think it has too small a convergence radius to go down to -1, due to near singularities in the complex plane, at least for the regular iteration (and I presume, as some experiments with continuum sum detailed here for the simple case of tetration converging to a fixed point, which currently seems to be the only handleable case, suggest, it "should" agree with the regular iteration), so what would one do if one wanted to compute it via the continuum sum formula with a Taylor series at 0 or -1, or is this impossible (or do we need some way to sum Taylor series outside the convergence radius... Mittag-Leffler again?)?

Possibly Related Threads…
Thread Author Replies Views Last Post
  f(x+y) g(f(x)f(y)) = f(x) + f(y) addition formula ? tommy1729 1 230 01/13/2023, 08:45 PM
Last Post: tommy1729
Question Formula for the Taylor Series for Tetration Catullus 8 2,568 06/12/2022, 07:32 AM
Last Post: JmsNxn
  There is a non recursive formula for T(x,k)? marraco 5 4,671 12/26/2020, 11:05 AM
Last Post: Gottfried
  Extrapolated Faá Di Bruno's Formula Xorter 1 5,383 11/19/2016, 02:37 PM
Last Post: Xorter
  Explicit formula for the tetration to base [tex]e^{1/e}[/tex]? mike3 1 6,375 02/13/2015, 02:26 PM
Last Post: Gottfried
  fractional iteration by schröder and by binomial-formula Gottfried 0 4,516 11/23/2011, 04:45 PM
Last Post: Gottfried
  simple base conversion formula for tetration JmsNxn 0 5,260 09/22/2011, 07:41 PM
Last Post: JmsNxn
  Change of base formula using logarithmic semi operators JmsNxn 4 13,786 07/08/2011, 08:28 PM
Last Post: JmsNxn
  Breaking New Ground In The Quest For The "Analytical" Formula For Tetration. mike3 5 15,053 05/09/2011, 05:08 AM
Last Post: mike3
  Constructing the "analytical" formula for tetration. mike3 13 32,197 02/10/2011, 07:35 AM
Last Post: mike3

Users browsing this thread: 1 Guest(s)