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
\( \hspace{24}
asp(x) =\sum_{h=0}^{\infty} (-1)^h U^{\circ h}(x) \)
and of increasing negative heights
\( \hspace{24}
asn(x) =\sum_{h=0}^{\infty} (-1)^h U^{\circ -h}(x) \)
then my conjecture, based on diagonalization was
\( \hspace{24} asp(x) + asn(x) - x =0 \)
which was wrong with a certain systematic error
\( \hspace{24} asp(x) + asn(x) - x =d(x) \)
I have now a description for the error d(x), which fits very well.
First, let's formally write \( \hspace8 as(x) = asp(x)+asn(x)-x \)as the two-way-infinite series
\( \hspace{24}
as(x) =\sum_{h=-\infty}^{\infty} (-1)^h*U^{\circ h}(x) \)
Then it is obvious, that as(x) is periodic with 2h, if x is expressed as powertower
\( \hspace{24} x =U^{\circ y}(1) = U^{\circ 2h+r}(1) \)
where h is integer and r is the fractional remainder of a number y (mod 2)
\( \hspace{24} x= U^{\circ 2h+r}(1) \)
and define
\( \hspace{24} 1_r= U^{\circ r}(1) \)
Then we may discuss as_r = as(x) as
\( \hspace{24} as_r = as(U^{\circ r}(1))=as(1_r) \) where r = y (mod 2)
We can then rewrite the formula
\( \hspace{24} asp(x)+asn(x)-x = d(x) \)
as
\( \hspace{24} asp_r + asn_r - 1_r = as_r = d_r \)
My observation was, that d_r is sinusoidal with r, with a very good fit (I checked also various bases t).
I got now
\( \hspace{24} as_r (=d_r) = a * \sin(r *\pi + w) \)
where "a" indicates the amplitude and "w" a phase-shift (depending on base t).
The following fits the result very well:
\( \hspace{24} a = \sqrt{ (as_0)^2 + (as_{0.5})^2)} \)
\( \hspace{24} w = \text{atan2}(as_0 , as_{0.5}) \)
so we could rewrite \( \hspace{24} asp_r + asn_r - 1_r \) as functional equation for asp
\( \hspace{24} asp_r = a * \sin(r*\pi + w) - (asn_r - 1_r) \)
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
\( \hspace{24}
asp(x) =\sum_{h=0}^{\infty} (-1)^h U^{\circ h}(x) \)
and of increasing negative heights
\( \hspace{24}
asn(x) =\sum_{h=0}^{\infty} (-1)^h U^{\circ -h}(x) \)
then my conjecture, based on diagonalization was
\( \hspace{24} asp(x) + asn(x) - x =0 \)
which was wrong with a certain systematic error
\( \hspace{24} asp(x) + asn(x) - x =d(x) \)
I have now a description for the error d(x), which fits very well.
First, let's formally write \( \hspace8 as(x) = asp(x)+asn(x)-x \)as the two-way-infinite series
\( \hspace{24}
as(x) =\sum_{h=-\infty}^{\infty} (-1)^h*U^{\circ h}(x) \)
Then it is obvious, that as(x) is periodic with 2h, if x is expressed as powertower
\( \hspace{24} x =U^{\circ y}(1) = U^{\circ 2h+r}(1) \)
where h is integer and r is the fractional remainder of a number y (mod 2)
\( \hspace{24} x= U^{\circ 2h+r}(1) \)
and define
\( \hspace{24} 1_r= U^{\circ r}(1) \)
Then we may discuss as_r = as(x) as
\( \hspace{24} as_r = as(U^{\circ r}(1))=as(1_r) \) where r = y (mod 2)
We can then rewrite the formula
\( \hspace{24} asp(x)+asn(x)-x = d(x) \)
as
\( \hspace{24} asp_r + asn_r - 1_r = as_r = d_r \)
My observation was, that d_r is sinusoidal with r, with a very good fit (I checked also various bases t).
I got now
\( \hspace{24} as_r (=d_r) = a * \sin(r *\pi + w) \)
where "a" indicates the amplitude and "w" a phase-shift (depending on base t).
The following fits the result very well:
\( \hspace{24} a = \sqrt{ (as_0)^2 + (as_{0.5})^2)} \)
\( \hspace{24} w = \text{atan2}(as_0 , as_{0.5}) \)
so we could rewrite \( \hspace{24} asp_r + asn_r - 1_r \) as functional equation for asp
\( \hspace{24} asp_r = a * \sin(r*\pi + w) - (asn_r - 1_r) \)
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.
Gottfried Helms, Kassel