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+--- Thread: Funny method of extending tetration? (/showthread.php?tid=527)
Funny method of extending tetration? - mike3 - 11/01/2010
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?
RE: Funny method of extending tetration? - Gottfried - 11/02/2010
(11/01/2010, 08:32 PM)mike3 Wrote: offset so it equals 1 at 0). Now, what doesHi Mike -
\( \lim_{k \rightarrow \infty} \mathrm{reg}_{\mathrm{RFP}}[e_{2k+1}^x](1) \)
do?
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