(10/22/2016, 10:29 PM)Vladimir Reshetnikov Wrote: I found an analytic function given by an explicit series, that generalizes the tetration in the sense that it reproduces its values for all positive integer heights and satisfies the same functional relation for all arguments in its domain of analyticity. Numerically it seems to match Kneser's solution, a PARI/GP program for which was posted on this forum. Any questions and critique are welcome. A short note explaining my solution is attached and is also available at https://tinyurl.com/tetration
Vladimir - I've looked into your *.pdf-announcement and see, that your method is -at least: related- to a concept which I call "iteration-series" (because it contains "iterates of x" instead of powers of x) and I am much interested to see more applications of that concept. It would be great if you could show the Pari/GP-implementation so I do not need to reengineer and possibly introduce bugs.
Gottfried
PS: moreover the denominators look suspiciously like a formula for the terms of the series which I derived by computing the powerseries for the Schröderfunction for some base/fixpoint (base in the Euler range) kept symbolically , see http://go.helms-net.de/math/tetdocs/Cont...ration.pdf Sec. 4.3, pg 24 (I use "t" for the fixpoint \( \;^\infty a \) and "u" for its logarithm there.)
But that is finally a power series with polynomial coefficients and not an iteration-series (with iteration-polynomials as coefficients) - I was much after that concept of iteration-series when I detected the well-known difficulties with that type of power series but couldn't make much progress beyond the so called "Newton"-series for fractional iterations ... (A bit on the Newton-series is here: http://go.helms-net.de/math/tetdocs/Bino...zation.htm )
Gottfried Helms, Kassel