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 \( f(x)=\exp(x)-1 \), 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

\( A_0 = I \) (the identity-matrix) and then \( A_{k+1}=U^{A_k} \) . Here the exponentiation is done by \( U^A = \exp(L*A) \) such that finally \( U^{A_\infty} = A_\infty \) . 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 \( g(x) = \sum a_k x^k \) 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 \( f(x)=\exp(x)-1 \), 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

\( A_0 = I \) (the identity-matrix) and then \( A_{k+1}=U^{A_k} \) . Here the exponentiation is done by \( U^A = \exp(L*A) \) such that finally \( U^{A_\infty} = A_\infty \) . 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 \( g(x) = \sum a_k x^k \) 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...

Gottfried Helms, Kassel