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 Transseries, nest-series, and other exotic series representations for tetration bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 11/29/2009, 09:38 AM (11/29/2009, 09:09 AM)Daniel Wrote: I can compute a nested series for the fractional iterates of $e^x-1$, but I don't claim the series converges. I think the series is a formal power series. It is interesting to know that the series is Borel-summable. So your Schröder sums compute the regular iteration, is that true? I think it is very important to know those equalities. For example it took a while until I realized that the matrix approach introduced by Gottfried is actually equal to the regular iteration. As a test, e.g. the regular half-iterate of $e^x-1$ has as the first 10 coefficients: $0$, $1$, $\frac{1}{4}$, $\frac{1}{48}$, $0$, $\frac{1}{3840}$, $-\frac{7}{92160}$, $\frac{1}{645120}$, $\frac{53}{3440640}$, $-\frac{281}{30965760}$ Or generally the $t$-th iterate has as the first 10 coefficients $0$, $1$, $\frac{1}{2} t$, $\frac{1}{4} t^{2} - \frac{1}{12} t$, $\frac{1}{8} t^{3} - \frac{5}{48} t^{2} + \frac{1}{48} t$, $\frac{1}{16} t^{4} - \frac{13}{144} t^{3} + \frac{1}{24} t^{2} - \frac{1}{180} t$, $\frac{1}{32} t^{5} - \frac{77}{1152} t^{4} + \frac{89}{1728} t^{3} - \frac{91}{5760} t^{2} + \frac{11}{8640} t$, $\frac{1}{64} t^{6} - \frac{29}{640} t^{5} + \frac{175}{3456} t^{4} - \frac{149}{5760} t^{3} + \frac{91}{17280} t^{2} - \frac{1}{6720} t$, $\frac{1}{128} t^{7} - \frac{223}{7680} t^{6} + \frac{1501}{34560} t^{5} - \frac{37}{1152} t^{4} + \frac{391}{34560} t^{3} - \frac{43}{32256} t^{2} - \frac{11}{241920} t$, $\frac{1}{256} t^{8} - \frac{481}{26880} t^{7} + \frac{2821}{82944} t^{6} - \frac{13943}{414720} t^{5} + \frac{725}{41472} t^{4} - \frac{2357}{580608} t^{3} + \frac{17}{107520} t^{2} + \frac{29}{1451520} t$, Does that match your findings? I think these formulas are completely derivable from integer-iteration. If one knows that each coefficient is just a polynomial then this polynomial is determined by the number of degree plus 1 values for $t$ and these can be gained by just so many consecutive integer values. So this sounds really like your Schröder summation. However an alternative approach is just to solve the equation $f^{\circ t}\circ f=f\circ f^{\circ t}$ for $f^{\circ t}$, where $f$ and $f^{\circ t}$ are treated as formal powerseries. « 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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