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Tetra-series
#15
Assume for the following a fixed base t for the dexp- or "U"-tetration, so that dxp°h(x):=dxp_t°h(x) and for shortness let me replace dxp by U in the following. Also I change some namings from the previous posts for consistency.

Denote the alternating series of U-powertowers of increasing positive heights

and of increasing negative heights

then my conjecture, based on diagonalization was



which was wrong with a certain systematic error



I have now a description for the error d(x), which fits very well.

First, let's formally write as the two-way-infinite series



Then it is obvious, that as(x) is periodic with 2h, if x is expressed as powertower

where h is integer and r is the fractional remainder of a number y (mod 2)



and define



Then we may discuss as_r = as(x) as

where r = y (mod 2)

We can then rewrite the formula



as



My observation was, that d_r is sinusoidal with r, with a very good fit (I checked also various bases t).

I got now



where "a" indicates the amplitude and "w" a phase-shift (depending on base t).

The following fits the result very well:




so we could rewrite as functional equation for asp



and also can determine all as_r using asn_r for r=0 and r=0.5 (the integer and half-integer-iterate U°0(1) and U°0.5(1)) only.

Note, that the computation of asp_r is exact using the appropriate matrix (I + Ut)^-1 of the diagonalization-method.
The diagonalization-method deviates only for the part asn_r; it gives asn_r - d_r instead of asn_r ; unfortunately, this does not allow to determine d_r correctly with the diagonalization-method only (yet).

The benefit of the diagonalization-method is here, that its matrix (I+Ut)^-1 provides the coefficients for a powerseries for asp_r, which seems to be the analytic continuation for bases t, where the series asp would diverge.
Gottfried Helms, Kassel
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Messages In This Thread
Tetra-series - by Gottfried - 11/20/2007, 12:47 PM
RE: Tetra-series - by andydude - 11/21/2007, 07:14 AM
RE: Tetra-series - by Gottfried - 11/22/2007, 07:04 AM
RE: Tetra-series - by andydude - 11/21/2007, 07:51 AM
RE: Tetra-series - by Gottfried - 11/21/2007, 09:41 AM
RE: Tetra-series - by Ivars - 11/21/2007, 03:58 PM
RE: Tetra-series - by Gottfried - 11/21/2007, 04:37 PM
RE: Tetra-series - by Gottfried - 11/21/2007, 06:59 PM
RE: Tetra-series - by andydude - 11/21/2007, 07:24 PM
RE: Tetra-series - by Gottfried - 11/21/2007, 07:49 PM
RE: Tetra-series - by andydude - 11/21/2007, 08:39 PM
RE: Tetra-series - by Gottfried - 11/23/2007, 10:47 AM
RE: Tetra-series - by Gottfried - 12/26/2007, 07:39 PM
RE: Tetra-series - by Gottfried - 02/18/2008, 07:19 PM
RE: Tetra-series - by Gottfried - 06/13/2008, 07:15 AM
RE: Tetra-series - by Gottfried - 06/22/2008, 05:25 PM
Tetra-series / Inverse - by Gottfried - 06/29/2008, 09:41 PM
RE: Tetra-series / Inverse - by Gottfried - 06/30/2008, 12:11 PM
RE: Tetra-series / Inverse - by Gottfried - 07/02/2008, 11:01 AM
RE: Tetra-series / Inverse - by andydude - 10/31/2009, 10:38 AM
RE: Tetra-series / Inverse - by andydude - 10/31/2009, 11:01 AM
RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 01:25 PM
RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 02:40 PM
RE: Tetra-series / Inverse - by andydude - 10/31/2009, 09:37 PM
RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 10:33 PM
RE: Tetra-series / Inverse - by Gottfried - 11/01/2009, 07:45 AM
RE: Tetra-series / Inverse - by andydude - 11/03/2009, 03:56 AM
RE: Tetra-series / Inverse - by andydude - 11/03/2009, 04:12 AM
RE: Tetra-series / Inverse - by andydude - 11/03/2009, 05:04 AM
RE: Tetra-series / Inverse - by Gottfried - 10/31/2009, 12:58 PM

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