(07/15/2022, 10:03 AM)Gottfried Wrote: Hmm, one more comment, why the property \( f°^{a+b}(z) = f°^b(f°^a(z)) \) holds for \( f(z) = \sin(z) \)Thank you for your posting Gottfried. I will be giving your material a close reading and repeated reading. I have long felt that our work is similar, you just attack iteration using matrices. I am fascinated by your MO postings on the fractional iteration of the sin function! We derive the same \(sin^n(z)\)! Very cool!!!
The Carleman-ansatz gives a triangular Carlemanmatrix, say \( S \) for the sine-function. \( S \) however has the diagonal of \( 1 \), so the diagonalization (which were then the operationalizing of the Schroeder-mechanism) cannot be applied here.
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See also my answer1 and/or answer2 in MO where this is a bit described and in which complete sequence of answers are many more valuable informations.
Gottfried
Gottfried, do you have any feel for which techniques discussed and developed here give equivalent results, at least for the sin function?
Daniel