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
 =\sum_{h=0}^{\infty} (-1)^h U^{\circ h}(x) )
and of increasing negative heights
 =\sum_{h=0}^{\infty} (-1)^h U^{\circ -h}(x) )
then my conjecture, based on diagonalization was
 + asn(x) - x =0 )
which was wrong with a certain systematic error
 + asn(x) - x =d(x) )
I have now a description for the error d(x), which fits very well.
First, let's formally write = asp(x)+asn(x)-x )
as the two-way-infinite series
 =\sum_{h=-\infty}^{\infty} (-1)^h*U^{\circ h}(x) )
Then it is obvious, that as(x) is periodic with 2h, if x is expressed as powertower
 = U^{\circ 2h+r}(1) )
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
)=as(1_r) )
where r = y (mod 2)
We can then rewrite the formula
+asn(x)-x = d(x))
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
 = a * \sin(r *\pi + w) )
where "a" indicates the amplitude and "w" a phase-shift (depending on base t).
The following fits the result very well:
^2 + (as_{0.5})^2)} )
 )
so we could rewrite
as functional equation for asp
 - (asn_r - 1_r) )
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
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
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
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
