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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 11/28/2009, 06:36 AM (This post was last modified: 11/28/2009, 06:37 AM by mike3.) (11/28/2009, 04:56 AM)andydude Wrote: What are "magic" coefficients? On this page: http://eom.springer.de/s/s087230.htm there's a formula for the "Mittag-Leffler expansion in a star", which is not a Taylor series, but a different type of series that is a sum of polynomials that converges over a whole star (it's explained on the page -- and contrast this with a Taylor series which only converges in a circle when the function is not entire). It looks like a two nested sums: $f(z) = \sum_{n=0}^{\infty} \sum_{\nu=0}^{k_n} c_{\nu}^{(n)} \frac{f^{(\nu)}(a)}{\nu!} (z - a)^{\nu}$ (and is a special case of the "nested series" and "transseries" I mention in the thread title) The "magic" numbers are the polynomial degrees $k_n$ and the coefficients $c_{\nu}^{(n)}$ on the terms. According to the site these are "independent of the form of $f(z)$ and can be evaluated once and for all", yet how to do this is not explained. « 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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