Understanding Abel/Schroeder with matrix-expression

[Gottfried]

Hmm, up to now I never could get a grip about the problems of the Abel-function. I merely took it as a notational formalism - perhaps the eigen-decomposition and your previous statements here helps me, to get a first step to such a grip now.

If I write

Code:

´
   V(x)~ * Bb^h = V(y)~

where \( y = \exp_b^{\circ h}(x) \) in Tex (or y = x {4,b} h in my partisan notation ), then writing

Code:

´
   t = h(b) which makes t^(1/t) =b
   u = log(t)

and the diagonalization (I should better write W_u, since W_u is dependent on u but use W for shortness here)

Code:

´
    Bb   = W * dV(u)   *W^-1

    Bb^h = W * dV(u)^h *W^-1
         = W * dV(u^h) *W^-1

then the second column of W provides the coefficients of a powerseries, which defines a function wu(x)

Code:

´
    wu(x) = V(x)~*W [,1]     // [,1] means second column of W
          = sum{k=0,inf} W[k,1]*x^k

then

Code:

´
    V(x)~ * W * dV(u^h) * W^-1 = V(y)~
    V(x)~ * W * dV(u^h)        = V(y)~ * W

and from dV(u^h) being diagonal it follows (and it suffices to use column 1 only)

Code:

´
(V(x)~ * W * dV(u^h)) [,1]  = (V(y)~ * W) [,1]
(V(x)~ * W [,1])* u^h        = V(y)~ * W  [,1]

and then the functional relation

Code:

´
wu(x)* u^h = wu(y)

from where , writing lg_b(x) = log(x)/log(b)

Code:

´
lg_u(wu(x)) + h = lg_u(wu(y))
or
h = lg_u(wu(y)) - lg_u(wu(x))

as I also noted recently in the older Abel-thread.

Isn't then lg_u(wu(x)) such an Abel-function?

[update] well, I've several times read the post of Andrew (or -another instance- of Henryk) which I never understood (and felt being left just to skim through after few lines, leaving it "at the expert's level") because I could not resolve the many unknowns into a system, which I'd been able to relate to my handwaved approach. But rereading it now, it seems to me, the above relation is identical to that considerations. At least the formal handling says this to me. Shame on me... [/update]
[some more edit-updates in the original part above]