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 Tensor power series andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 05/22/2008, 04:36 AM (This post was last modified: 05/22/2008, 04:39 AM by andydude.) I just realized I never explained how that function is related to the exponential factorial. The exponential factorial is defined as $EF(x) = x^{EF(x-1)}$ where $EF(1) = 1$, which is a univariate function. To make this function from the function above, you can first notice the pattern in its iterates: $ \begin{tabular}{rl} E^{\circ 1}(c, 1) = E(c, 1) & = (1, 2) \\ E^{\circ 2}(c, 1) = E(1, 2) & = (2^1, 3) \\ E^{\circ 3}(c, 1) = E(2, 3) & = (3^{2^1}, 4) \\ E^{\circ 4}(c, 1) = E(9, 4) & = (4^{3^{2^1}}, 5) \end{tabular}$ where c could theoretically be anything, since $1^c = 1$. And since I haven't indicated whether we are using 0-based or 1-based vectors, I'm going to use another contraction to get the top value: $ EF(x) = \left[{1 \atop 0}\right] \cd E^{\circ x}\left[{c \atop 1}\right]$ Now it should be obvious from this that $EF(0) = c$ which begs the question in my Conjectures post: what is EF(0)? Andrew Robbins « Next Oldest | Next Newest »

 Messages In This Thread Tensor power series - by andydude - 05/13/2008, 07:58 AM RE: Tensor power series - by andydude - 05/13/2008, 07:59 AM RE: Tensor power series - by andydude - 05/13/2008, 08:11 AM RE: Tensor power series - by andydude - 05/14/2008, 06:18 AM RE: Tensor power series - by Gottfried - 05/20/2008, 08:39 PM RE: Tensor power series - by andydude - 05/22/2008, 12:58 AM RE: Tensor power series - by andydude - 05/22/2008, 04:11 AM RE: Tensor power series - by andydude - 05/22/2008, 04:36 AM RE: Tensor power series - by bo198214 - 05/24/2008, 10:10 AM RE: Tensor power series - by andydude - 06/04/2008, 08:08 AM

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