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Funny method of extending tetration?

Does this lead anywhere, or does it just fail?

Take the partial sums of , a function called the "exponential sum function":


If we take this at odd values of , there will be a real fixed point. Take the regular iteration at this real fixed point, shifted so that it equals 1 at 0, call it , that is, regular iteration developed at the real fixed point, iterating "1" (i.e. offset so it equals 1 at 0). Now, what does

(11/01/2010, 08:32 PM)mike3 Wrote: offset so it equals 1 at 0). Now, what does

Hi Mike -

Hmm I tried with n=5,n=17,n=27 and got the according Bell-matrices.
Using integer heights iteration worked as expected.
I tried fractional heights, h=0.5,h=0.25 at x=0 and x=1 and the series with truncation at 64 coefficients do not converge well, not even having alternating signs.
So I couldn't discern any interesting result so far... Would you mind to give some more hint?

Gottfried Helms, Kassel

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