Infinite tetration of the imaginary unit

[Gottfried]

[ "... By the formulas in question, it is possible to determine the coefficients of the formal power series of \( h_f \); the denominators of these coefficients can be written as products of the form \( \lambda^n-\lambda \), for n>=2, since \( \alpha \) is an irrational number these products could be very small....", which is where the Brjuno number comes from.

Hi Sheldon -

in the thesis-article on page 2 and a bit more explicite on page 6 there is that formula for the rational approximation/irrationality measure of irrational numbers. Here the parameter mu is used and if I understand the argument correctly, then it says, if it is greater than 2 then there is some solution. Since all algebraic irrationals (roots of polynomials) have exactly the degree mu=2 the numbers mu>2 are all transcendental. That's also the background, on which Liouville-numbers may be of special interest: they have mu=infinite and are near the rational numbers again (just from "the other side of the globe", so-to-speak).

Unfortunately I am far too unfamiliar with all the higher concepts involved (even of the fourier-decomposition). So I cannot say much more than that the coefficients of h seem simply to satisfy the formula for the eigenmatrix/schroeder-function computed by diagonalization of a triangular Bell-matrix (for functions satisfying f(0)=0, and f'(0)<>0,1 ) - but that's only assumed by the appearance of the formula, I didn't look at it really deep.

Gottfried