[AIS] (alternating) Iteration series: Half-iterate using the AIS?

[Gottfried]

(...) I was curious which of the two superfunctions Gottfried's current algorithm was closer too. It is also possible to have a hybrid of these two superfunctions, which I posted on here, http://math.eretrandre.org/tetrationforu...hp?tid=515.
- Sheldon

Hi Sheldon -

what I found using the Bell-matrices for the lower and the upper fixpoint is, that the current method combines the results for the alternating iteration series developed at that fixpoints.
Say, the Bellmatrix B0 for the fixpoint-development at fp0=0 gives using \( AS_0 = (I + B_0)^{-1} \) the coefficients for a power series to determine the alternating iteration series beginning at x0 towards fp0 , let's call it asp(x) , correctly. But not towards the other direction (!). On the other hand, the Bellmatrix B1 for the fixpoint-development at the higher fixpoint fp1 gives using \( AS_1 = (I + B_1^{-1})^{-1} \) the coefficients for a power series to determine the alternating iteration series beginning at x0 towards fp1 , call it asn(x) .
Then the empirical observation that indeed asp(x)+asn(x)-x = asum(x) shows, that the asum(x) (which is taken by the two-way infinite iteration series ignoring the Bell-matrices with their fixpoint-specificity) is sort of "hybrid", ... and the fractional iteration based on it should then as well be taken as "hybrid" of the developments at the two different fixpoints (where one should explicite the concrete meaning of the "hybrid"ity into a formalized algebraic description).

Perhaps it would be fruitful to explicate this much more. Possibly I can provide some examples tomorrow. Also, if someone needs ready-made Pari/GP-procedures to check this him/herself, I can provide my current function-definitions.

Gottfried