(06/18/2022, 07:22 AM)Gottfried Wrote:(06/18/2022, 04:29 AM)Daniel Wrote: My work is experimentally consistent with Bell matrices. Are Bell matrices considered along with Carlemann matrices in people's research?
The Carleman-matrix is simply a factorially similarity scaling of the Bell matrix, and transposed. The idea is to work on the coefficients of the taylor-series, and so the work is practically identical between Carleman- and Bell-notation.
But note: it provides its concise arithmetic/algebra on *taylor-series* only, not, for instance, laurent series - and for analysis and bulding a "toolbox" for many prominent problems, you need
the inclusion of \(c_{-1} \cdot x^{-1}\) terms.
The only work with such an extension of the Carleman-ansatz I have ever seen is an article of Eri Jabotinsky, and when I read it it was over my head, and I had only spurious time for this matter then, so I did not step in. However I think it is a serious AND required extension of the range of applicability, it makes the Carleman-ansatz more useful and then surely wider known.
Gottfried
An exposition "from the ground" is in ContinuousfunctionalIteration (at my webspace for tetration) ; I think you'll find in it many things known to yourself (for instance the description in terms of derivatives) but I wrote this for the beginner as well as for a self-reflection of my knowledge.
I'd like to add, that Carleman matrices are guaranteed to exist. So, Kneser doesn't prove that they exist. Kneser proves there is a certain Riemann mapping which makes a solution to tetration. But, in and of itself, this means we can use Carleman matrices to find a solution. So, Carleman matrix method will work; there is some sequence of matrices that does work. But if you try to construct Kneser from scratch using Carleman; then you have a very different problem.
This is where I shy away from Carleman... It's like trying to start a fire with sticks and stones, when rocketfuel is right next door.