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 Funny method of extending tetration? mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 11/01/2010, 08:32 PM Hi. Does this lead anywhere, or does it just fail? Take the partial sums of $\exp$, a function called the "exponential sum function": $e_n(x) = \sum_{i=0}^{n} \frac{x^i}{i!}$. If we take this at odd values of $n$, 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 $\mathrm{reg}_{\mathrm{RFP}}[e_n^x](1)$, that is, regular iteration developed at the real fixed point, iterating "1" (i.e. offset so it equals 1 at 0). Now, what does $\lim_{k \rightarrow \infty} \mathrm{reg}_{\mathrm{RFP}}[e_{2k+1}^x](1)$ do? Gottfried Ultimate Fellow Posts: 789 Threads: 121 Joined: Aug 2007 11/02/2010, 12:20 PM (11/01/2010, 08:32 PM)mike3 Wrote: offset so it equals 1 at 0). Now, what does $\lim_{k \rightarrow \infty} \mathrm{reg}_{\mathrm{RFP}}[e_{2k+1}^x](1)$ do?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 Gottfried Helms, Kassel « Next Oldest | Next Newest »

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