generalizing the problem of fractional analytic Ackermann functions

[Gottfried]

That's a good way to approach the question. I'm not too familiar with carleman matrices but I think it goes something like this

(...)

Hi James -

unfortunately I can't follow in the above. But concerning tetration of carleman-matrices themselves there is a simple example: the pascal-matrix is the carleman-matrix for the operation of incrementation by 1 and powers of it are carleman-matrices for the operation of addition. I've worked out an example how to tetrate the pascal-matrix, perhaps this is useful (and possibly generalizable). See http://go.helms-net.de/math/tetdocs/index.htm go to the short statement at "pascal matrix tetrated" and open http://go.helms-net.de/math/tetdocs/Pasc...trated.pdf

Nearly all what I'm doing in tetration is based on this concept of "carleman-matrices" and I might be able to answer if you have some concrete questions. I did not know that name when I stumbled on that concept so my first expositions of my fiddlings are mostly in threads headed by "matrix-method" and might be informative/helpful before (though it is extremely exploratory and not well structured from the beginning)

Gottfried