Transseries, nest-series, and other exotic series representations for tetration

[mike3]

What are "magic" coefficients?

On this page:

http://eom.springer.de/s/s087230.htm

there's a formula for the "Mittag-Leffler expansion in a star", which is not a Taylor series, but a different type of series that is a sum of polynomials that converges over a whole star (it's explained on the page -- and contrast this with a Taylor series which only converges in a circle when the function is not entire). It looks like a two nested sums:

\( f(z) = \sum_{n=0}^{\infty} \sum_{\nu=0}^{k_n} c_{\nu}^{(n)} \frac{f^{(\nu)}(a)}{\nu!} (z - a)^{\nu} \)

(and is a special case of the "nested series" and "transseries" I mention in the thread title)

The "magic" numbers are the polynomial degrees \( k_n \) and the coefficients \( c_{\nu}^{(n)} \) on the terms. According to the site these are "independent of the form of \( f(z) \) and can be evaluated once and for all", yet how to do this is not explained.