A sort of mind-breaking...
I tried to step into the subject of Abel-/Schroeder-function from the functional approach, leaving the matrix-considerations aside, and stepped into a problem.
If I assume
\( \hspace{24} f_b(x) = b^x \)
\( \hspace{24} f_b^{^oh}(x) = b^{...^{b^x}} \) (b occurs height h-times)
replaced by a b-related function pb(x) and its inverse qb(x) such that
\( \hspace{24} f_b^{^oh}(x)= q_b( u^h*p_b(x)) \)
(as a sketch only - leaving for instance the fixpoint-shift question aside) and I assume \( p_b(x) \) and \( q_b(x) \) as b-parametrized powerseries in x (and "u" as the base of the assumed eigenvalues of the diagonalisation) then I observe the problem, that the scalar product \( y = u^h*p_b(x) \), where \( p_b(x) \) is a constant "c" for a chosen "x" and only "h" is variable, is independent of the order of its factors; so the function \( q_b(y) \) "does not know" about the order of the "u"'s and "c". But anyway we expect the effect of "c" always on top of the powertower, which clearly implies some ordering.
From this I question, whether we are correct in assuming the existence of a *scalar* function \( p_b(x) \). Don't we need a notion of an object, which assures, that with different orders in \( u*u*u*\ldots *u*p_b(x) \) the expression has different values?
The matrix-notation gives this order implicitely, but a scalar expression like the above doesn't depend on any order...
Gottfried
I tried to step into the subject of Abel-/Schroeder-function from the functional approach, leaving the matrix-considerations aside, and stepped into a problem.
If I assume
\( \hspace{24} f_b(x) = b^x \)
\( \hspace{24} f_b^{^oh}(x) = b^{...^{b^x}} \) (b occurs height h-times)
replaced by a b-related function pb(x) and its inverse qb(x) such that
\( \hspace{24} f_b^{^oh}(x)= q_b( u^h*p_b(x)) \)
(as a sketch only - leaving for instance the fixpoint-shift question aside) and I assume \( p_b(x) \) and \( q_b(x) \) as b-parametrized powerseries in x (and "u" as the base of the assumed eigenvalues of the diagonalisation) then I observe the problem, that the scalar product \( y = u^h*p_b(x) \), where \( p_b(x) \) is a constant "c" for a chosen "x" and only "h" is variable, is independent of the order of its factors; so the function \( q_b(y) \) "does not know" about the order of the "u"'s and "c". But anyway we expect the effect of "c" always on top of the powertower, which clearly implies some ordering.
From this I question, whether we are correct in assuming the existence of a *scalar* function \( p_b(x) \). Don't we need a notion of an object, which assures, that with different orders in \( u*u*u*\ldots *u*p_b(x) \) the expression has different values?
The matrix-notation gives this order implicitely, but a scalar expression like the above doesn't depend on any order...
Gottfried
Gottfried Helms, Kassel