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Regular iteration using matrix-Jordan-form
(12/12/2013, 09:44 AM)sheldonison Wrote: Any idea how this is related to the fractional iteration of exp(x)-1? I couldn't figure anything out from the thread, or the link.
- Sheldon
I've looked into the book of L. Comtet, from whom this whole sequence originated. Hmm, either me or the text seemed a bit obfuscated, but I've not yet understood, how he used this matrix of values; he talks about Faa di Bruno-Formula, but the whole argumentation/path to the fractional iterate is somehow complicated.(It's on the pages 144-148 in the"Advanced combinatorics" (Reidel, Dordrecht)) - I could provide a page scan if needed.

But after I see, that this set of polynomials perfectly agrees with the matrix of the Jordan-decomposition, which in the case of the diagonalizable Carlemanmatrix is the eigenmatrix and gives the coefficients of the Schröder-function, I think, it is a simple analogy here: it might simply implement a Schröder-function for the non-diagonalizable case for tha base b=exp(1) and the function f(x) = b^x - 1 .

Why I draw attention to this here is, because the method to use Jordandecomposition for nondiagonalizeble Bell/Carleman-matrices might be a very basic systematic extension for many related iteration-problems. I think, for instance, E Schröder could have come up with that generalization already, but Jordan decomposition might have seen to a very difficult procedure to be calculated and one does not find many references in the mass-literature. Even contemporary examples: while the for Pascal-matrix and their powers etc, which is frequently discussed even in matrix-contexts I could only find one article which studies Jordan-decomposition of the Pascal-matrix; but none for the same using the Stirling matrix - although one would arrive at the Comtet-numbers just completely natural...

Gottfried Helms, Kassel

Messages In This Thread
RE: [MO]: regular iteration using matrix-Jordan-form (a surprising observation) - by Gottfried - 12/12/2013, 10:52 AM

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