Here I'm considering fractional heights h for the function Ut°h(x) of complex base t, which implemets x -> t^x -1 and hope to get to the source of the persisting problem.
Note, that with the previously described method everything is fine with integral heights, so the problem should be somehow systematic.
Remember the Schröder-function, whose powerseries-coefficients I find in the 2'nd column of the eigenvectormatrix of the operator Ut, such that
\( \hspace{24} U_t = W * ^dV(u) * WI \)
where u=log(t) and WI is the inverse of W.
Then the Schröder-function sigma for parameter t is
\( \hspace{24} \sigma_t(x) = \sum_{k=0}^{\infty} W_{k,1}*x^k \)
and the inverse takes the coefficients of WI instead:
\( \hspace{24} \sigma_t^{-1}(x) = \sum_{k=0}^{\infty} WI_{k,1}*x^k \)
To compute Ut°1(x) we simply do
\( \hspace{24} U_t^{{^o}1}(x) = \sigma_t^{-1}(u*\sigma_t(x)) \)
or, for integer h
\( \hspace{24} U_t^{{^o}h}(x) = \sigma_t^{-1}(u^h*\sigma_t(x)) \)
As I said, this works fine even for complex t, checked with some t where abs(t)<1, if heights are integral.
But we'll see, that for fractional h this does not work.
With this simple process, for instance t=1/2 + 1/2 I there is also no severe problem with convergence or even divergence - things sum up in a completely easy manner - but anyway, the fractional h in u^h produce obviously complete crap.
Currently I suspect, that there must be some conjugacy be involved - but don't have a working idea, where.
Note, that with the previously described method everything is fine with integral heights, so the problem should be somehow systematic.
Remember the Schröder-function, whose powerseries-coefficients I find in the 2'nd column of the eigenvectormatrix of the operator Ut, such that
\( \hspace{24} U_t = W * ^dV(u) * WI \)
where u=log(t) and WI is the inverse of W.
Then the Schröder-function sigma for parameter t is
\( \hspace{24} \sigma_t(x) = \sum_{k=0}^{\infty} W_{k,1}*x^k \)
and the inverse takes the coefficients of WI instead:
\( \hspace{24} \sigma_t^{-1}(x) = \sum_{k=0}^{\infty} WI_{k,1}*x^k \)
To compute Ut°1(x) we simply do
\( \hspace{24} U_t^{{^o}1}(x) = \sigma_t^{-1}(u*\sigma_t(x)) \)
or, for integer h
\( \hspace{24} U_t^{{^o}h}(x) = \sigma_t^{-1}(u^h*\sigma_t(x)) \)
As I said, this works fine even for complex t, checked with some t where abs(t)<1, if heights are integral.
But we'll see, that for fractional h this does not work.
With this simple process, for instance t=1/2 + 1/2 I there is also no severe problem with convergence or even divergence - things sum up in a completely easy manner - but anyway, the fractional h in u^h produce obviously complete crap.
Currently I suspect, that there must be some conjugacy be involved - but don't have a working idea, where.
Gottfried Helms, Kassel