A couple of years ago I've fiddled with the question of the iterated exponentiation of the most simple(?) Carleman-matrix, that for the function f(x)=a+x, namely the Pascal-matrix (or the a'th power of it).
I have then found some interesting properties (see http://go.helms-net.de/math/tetdocs/Pasc...trated.pdf for the interested reader) , but besides of a still relatively vague idea it is difficult for me to say precisiely, what such an operation means and what it means, if then the tetrated pascal-matrix is used as provider of coefficients of some new associated operation.
Just playing around I've done this with the Carlemanmatrix for the=\exp(x)-1 )
, which is lower triangular, has units in the diagonal and for which a (matrix-) logarithm can be computed which can then be taken also as valid truncation of the infinite-sized-matrix.
Let's call the Carlemanmatrix for f(x) as "U", and its (matrix-)logarithm as "L" then the infinite exponentiation
(the identity-matrix) and then 
. Here the exponentiation is done by  )
such that finally 
. This occurs at the n'th step of iteration because the matrices L*A are nilpotent to the chosen matrix size nxn.
This gives then coefficients for some function which I would like to characterize. Maybe it is Pentation, but I'm not sure about this.
Anyway, the coefficients in the second column of A are strongly diverging, more than the factorial and it seems they are all positive, so that to be able to evaluate it at all in terms of a function = \sum a_k x^k )
requires at least a negative x, and also sophisticated methods for the divergent summation.
Before I begin to invest much time and energy in this I would like to have an idea, what such function g(x) would do, how it could be characterized, at least qualitatively...
Gottfried
p.s.: if someone interested in this needs the Pari/GP-code I can provide this. Please consider, that it will be work to flesh out the relevant procedures from my (slightly unstructured ;-)) collection of Pari/GP code samples, so I'd like to do this if there is seriously interest only...
I have then found some interesting properties (see http://go.helms-net.de/math/tetdocs/Pasc...trated.pdf for the interested reader) , but besides of a still relatively vague idea it is difficult for me to say precisiely, what such an operation means and what it means, if then the tetrated pascal-matrix is used as provider of coefficients of some new associated operation.
Just playing around I've done this with the Carlemanmatrix for the
Let's call the Carlemanmatrix for f(x) as "U", and its (matrix-)logarithm as "L" then the infinite exponentiation
This gives then coefficients for some function which I would like to characterize. Maybe it is Pentation, but I'm not sure about this.
Anyway, the coefficients in the second column of A are strongly diverging, more than the factorial and it seems they are all positive, so that to be able to evaluate it at all in terms of a function
Before I begin to invest much time and energy in this I would like to have an idea, what such function g(x) would do, how it could be characterized, at least qualitatively...
Gottfried
p.s.: if someone interested in this needs the Pari/GP-code I can provide this. Please consider, that it will be work to flesh out the relevant procedures from my (slightly unstructured ;-)) collection of Pari/GP code samples, so I'd like to do this if there is seriously interest only...
Gottfried Helms, Kassel