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 Superroots and a generalization for the Lambert-W andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 12/30/2015, 09:49 AM I believe I may have found a closed form for the power series of the third tetrate function as well. I'm not sure if these are known, but I just used the elementary properties of binomials and Stirling numbers to derive these: $ \begin{equation} {}^{3}x = \sum_{k=0}^{\infty} \log(x)^k \sum_{j=0}^{k} \sum_{i=0}^{k - j - 1} \frac{(k - j - i)^j j^i}{(k - j - i)!j!i!} \end{equation}$ $ \begin{equation} {}^{3}x = \sum_{k=0}^{\infty} (x - 1)^k \sum_{j=0}^{k} \sum_{J=0}^{j} \sum_{i=0}^{k} \sum_{I=0}^{i} {\left[{i \atop I}\right]} {\left[{j \atop J}\right]} {\left({J \atop {k - j - i}}\right)} \frac{J^I}{j!i!} \end{equation}$ The first one (logarithmic power series) reminds me of something in one of Galidakis' papers about tetration, but I don't remember which paper. The second one is derived from the fact that the generating function of the signed Stirling numbers the first kind is $(1 + x)^z$. « Next Oldest | Next Newest »

 Messages In This Thread Superroots and a generalization for the Lambert-W - by Gottfried - 11/09/2015, 01:17 PM RE: Superroots and a generalization for the Lambert-W - by nuninho1980 - 11/09/2015, 11:27 PM RE: Superroots and a generalization for the Lambert-W - by Gottfried - 11/10/2015, 12:06 PM RE: Superroots and a generalization for the Lambert-W - by tommy1729 - 11/10/2015, 09:38 AM RE: Superroots and a generalization for the Lambert-W - by Gottfried - 11/10/2015, 12:04 PM RE: Superroots and a generalization for the Lambert-W - by tommy1729 - 11/10/2015, 11:19 PM RE: Superroots and a generalization for the Lambert-W - by andydude - 11/13/2015, 05:58 PM RE: Superroots and a generalization for the Lambert-W - by Gottfried - 11/13/2015, 07:05 PM RE: Superroots and a generalization for the Lambert-W - by Gottfried - 11/16/2015, 01:08 AM RE: Superroots and a generalization for the Lambert-W - by andydude - 11/21/2015, 05:05 AM RE: Superroots and a generalization for the Lambert-W - by andydude - 11/22/2015, 08:12 PM RE: Superroots and a generalization for the Lambert-W - by andydude - 11/24/2015, 12:51 AM RE: Superroots and a generalization for the Lambert-W - by Gottfried - 11/24/2015, 02:56 AM RE: Superroots and a generalization for the Lambert-W - by andydude - 11/24/2015, 07:16 AM RE: Superroots and a generalization for the Lambert-W - by andydude - 11/24/2015, 07:00 AM RE: Superroots and a generalization for the Lambert-W - by Gottfried - 12/01/2015, 03:13 PM RE: Superroots and a generalization for the Lambert-W - by tommy1729 - 12/01/2015, 11:58 PM RE: Superroots and a generalization for the Lambert-W - by Gottfried - 12/02/2015, 03:49 AM RE: Superroots and a generalization for the Lambert-W - by tommy1729 - 12/02/2015, 01:22 PM RE: Superroots and a generalization for the Lambert-W - by andydude - 12/02/2015, 12:48 AM RE: Superroots and a generalization for the Lambert-W - by Gottfried - 12/02/2015, 02:43 AM RE: Superroots and a generalization for the Lambert-W - by andydude - 12/09/2015, 06:34 AM RE: Superroots and a generalization for the Lambert-W - by andydude - 12/30/2015, 09:49 AM

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