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 Transseries, nest-series, and other exotic series representations for tetration mike3 Long Time Fellow Posts: 368 Threads: 44 Joined: Sep 2009 12/15/2009, 06:56 AM Hmm. I think I may have figured out what's going on. It seems like Faulhaber's formula fails to sum $\sum_{n=0}^{x-1} e^{qn}$ for what appears to be $q \ge 2\pi$. Below that it works and sums it to, apparently $\frac{e^{qx} - 1}{e^q - 1}$. So that would provide an analytic continuation to higher q-values, thus the formula given is valid in a sense, but direct application of the Faulhaber formula will not work. It seems that a combination of analytic continuation and divergent summation theory applied to Faulhaber and transseries theory is needed to put together a comprehensive theory of continuum sums of general analytic functions. I find the appearance of $2\pi$ interesting, considering it is the magnitude of the imaginary period of the exponential function. I don't know if that is significant or not. « Next Oldest | Next Newest »

 Messages In This Thread Transseries, nest-series, and other exotic series representations for tetration - by mike3 - 11/26/2009, 09:46 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by Daniel - 11/26/2009, 03:57 PM RE: Transseries, nest-series, and other exotic series representations for tetration - by bo198214 - 11/26/2009, 04:42 PM RE: Transseries, nest-series, and other exotic series representations for tetration - by Daniel - 11/29/2009, 09:09 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by bo198214 - 11/29/2009, 09:38 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by Daniel - 12/01/2009, 02:56 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by bo198214 - 12/01/2009, 09:08 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by tommy1729 - 12/01/2009, 10:22 PM RE: Transseries, nest-series, and other exotic series representations for tetration - by mike3 - 11/27/2009, 01:29 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by andydude - 11/28/2009, 04:56 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by mike3 - 11/28/2009, 06:36 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by mike3 - 11/28/2009, 06:50 AM RE: Transseries, nest-series, and other exotic series representations for tetration - by kobi_78 - 12/14/2009, 07:17 PM

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