Question about tetration methods

[Gottfried]

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As Gottfried spoke, he meant that he can think of less than 5 solutions to tetration which are intrinsically unique. This is true. There is Kneser. There is Sheldon. There is Carlemann (programmed by Gottfried). There is Kouznetsov. There is Paulsen and Cowgill.

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Thanks, James, for your insightful msg!        



I just get the feeling, the ("Carleman-") matrix-approach is sometimes taken with a slight mystique expectations around it. So I'll reiterate some explanation (about which you might well be aware of, but likely not all other readers around).                             



Basically the matrix-approach is nothing else than a notational framework for the manipulation of power series (in terms of their coefficients). For instance, the Schröder-procedere for iteration of analytic functions having powerseries \(f(x)= \sum_{k=1}^\infty a_k x^k\) can be coded by appropriate matrix-operations, involving such "Carleman"-matrices, or we may say, that what we have to do to write down the manipulations of power series to implement the Schröder-functionality can be concisely be denoted by the matrix-formulae.    

When I started my engagement with tetration I didn't know anything about Schröder-/Abelfunction, Kneser, and all this stuff, and as well I did not know the name "Bell", nor "Carleman" matrix; I just fiddled with the idea of inventing a method to operate on the coefficients of powerseries, their powers, and their iterations.           



My primary object has been the Pascal-matrix, where I observed, that that matrix mapped the function \(f_1(x)=x\) to \(g_1(x)=x+1\) and in the same one sweep as well \(f_2(x)=x^2\) to \(g_2(x)=(x+1)^2\) and so on, when just set in the most simple matrix-multiplication scheme. Of course, powers of the pascalmatrix P made then iterations of that maps (discussion-article), and fiddling the same thing with other functions I came then to \(f(x)=\exp(x)\) where then soon the contact with the tetration-forum happened.   Still I did not know about any of the earlier research in this directions, only after Andrew Robbins coined the name "Bell"-matrix (and/or "Carleman"matrix) it was that I got aware that this naive playing around with the coefficients of powerseries had been done before (and has nothing of magic, for instance the fractional iteration of the \(\exp(x)-1\) by manipulating its powerseries has been discussed in the book of Comtet down to the operations with the coefficients, only that Comtet did not introduce a full-fledged matrix-notation for that algebra which is involved). The rediscovery of the Schröder-function in the context of this matrix-algebraic formulae was then only to understand that the completely common folklore of matrix-eigensystem-decomposition simply gives the coefficients that Schröder(?) found the other way around for his function.                       



So, for instance, it seems to me that there is a misconception of this all when I read that "the existence of Carlemanmatrices is/must be proven" or the like... It even might be, that the Riemann-map can be formulated in that matrix-algebraic notation, I only can't say this: since I do not understand enough of this mapping at all; if it can be expressed in terms of building powerseries at all then it should be possible, and it is not needed that the occuring matrices are "Carleman", they might be "Vandermonde", or whatelse ever.



Hmm - a long comment, perhaps not needed. On the other hand, if this comment is useful for understanding anyway, I could perhaps inserrt something in the hyperop-wiki (or in a separate thread)...      



Btw. it seems there is a good journey underway here, this days - may the summer be fruitful and glorious :-) 



Gottfried