Hmm, that is a surprise.
I just checked the vague impression, that the asum(x) is just "in the middle" of the of the matrix-based versions of that alternating sum, when I use the fixpoint-0-Bell-matrix for that half of the alternating sum, whose terms go to the fixpoint-0, and the fixpoint-1-Bell-matrix for the other half of the alternating sum (whose terms go to the fixpoint 1).
Bingo!
It's so simple as this. So I can determine the asum(x) using the matrix-method (=Neumannseries of Carlemanmatrix) as well, which seemed to be impossible before.
I don't yet know what it means, that the interpolation to fractional heights by this method involves both real fixpoints - perhaps this defines a meaningful alternative to the situation in the regular tetration, where we have two different and seemingly unrelated fractional heights depending on the selection of the fixpoint.
But another aspect might be more interesting: the matrix-based method seems to be able to analytically continue that alternating iteration series to bases outside the Euler-range, as I'd discussed elsewhere already 2007 or 2008 (for a reminder: it was worked out that it is very likely that the matrix-method employed here is able to assign a reasonable value to iteration series of the type \( S(x) = x - e^x + e^{e^x} - e^{e^{e^x}} + \cdots \pm \cdots \) see http://go.helms-net.de/math/tetdocs/Iter...tion_1.htm ). So I've now to check whether the current discovery can also be used to extend the asum-based fractional tetration to other bases...
other relevant posts in the forum:
http://math.eretrandre.org/tetrationforu...php?tid=38
http://math.eretrandre.org/tetrationforu...hp?tid=692
[update 31.3.2015] link to the discussion of the analytic continuation on my tetration-pages added
I just checked the vague impression, that the asum(x) is just "in the middle" of the of the matrix-based versions of that alternating sum, when I use the fixpoint-0-Bell-matrix for that half of the alternating sum, whose terms go to the fixpoint-0, and the fixpoint-1-Bell-matrix for the other half of the alternating sum (whose terms go to the fixpoint 1).
Bingo!
It's so simple as this. So I can determine the asum(x) using the matrix-method (=Neumannseries of Carlemanmatrix) as well, which seemed to be impossible before.
I don't yet know what it means, that the interpolation to fractional heights by this method involves both real fixpoints - perhaps this defines a meaningful alternative to the situation in the regular tetration, where we have two different and seemingly unrelated fractional heights depending on the selection of the fixpoint.
But another aspect might be more interesting: the matrix-based method seems to be able to analytically continue that alternating iteration series to bases outside the Euler-range, as I'd discussed elsewhere already 2007 or 2008 (for a reminder: it was worked out that it is very likely that the matrix-method employed here is able to assign a reasonable value to iteration series of the type \( S(x) = x - e^x + e^{e^x} - e^{e^{e^x}} + \cdots \pm \cdots \) see http://go.helms-net.de/math/tetdocs/Iter...tion_1.htm ). So I've now to check whether the current discovery can also be used to extend the asum-based fractional tetration to other bases...
other relevant posts in the forum:
http://math.eretrandre.org/tetrationforu...php?tid=38
http://math.eretrandre.org/tetrationforu...hp?tid=692
[update 31.3.2015] link to the discussion of the analytic continuation on my tetration-pages added
Gottfried Helms, Kassel
