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.

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