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 holomorphic binary operators over naturals; generalized hyper operators JmsNxn Long Time Fellow Posts: 566 Threads: 94 Joined: Dec 2010 07/24/2012, 07:14 PM (This post was last modified: 07/24/2012, 11:37 PM by JmsNxn.) I think that is a clever way of writing recursion. You made a typographical mistake though. If we write: $\mathcal{L}_s(x, x \,\,\bigtriangleup_s\,\,y) = y$ then we have recursion as: $x \,\,\bigtriangleup_{s-1}\,\,y = x\,\,\bigtriangleup_s\,\,\mathcal{L}_s(x,y) + 1$ However. I am only concerned with when $\mathcal{L}_s(x, y) \in \mathbb{N}$. Since going to reals requires an universal extension of hyper operators to reals. But! I had another aha! $\mathcal{L}_s(x,y) = \prod_{n=0}^{\infty} \mathcal{L}_n(x,y)^{\zeta_n(s)} = (y-x)^{\zeta_0(s)} \cdot (\frac{y}{x})^{\zeta_1(s)} \cdot (\log_x(y))^{\zeta_2(s)} \cdot (\text{slog}_x(y))^{\zeta_3(s)}\cdot...$ $\zeta_n(s) = \frac{\sin(\pi (s-n))}{\pi (s-n)} \psi_n(s)$ for some undetermined $\psi$ This would actually allow us to extend to some reals. Thanks for this formation of recursion. I think it's a more efficient formula. « Next Oldest | Next Newest »

 Messages In This Thread holomorphic binary operators over naturals; generalized hyper operators - by JmsNxn - 07/19/2012, 04:44 PM RE: holomorphic binary operators over naturals; generalized hyper operators - by JmsNxn - 07/19/2012, 05:49 PM RE: holomorphic binary operators over naturals; generalized hyper operators - by tommy1729 - 07/19/2012, 10:16 PM RE: holomorphic binary operators over naturals; generalized hyper operators - by JmsNxn - 07/19/2012, 10:26 PM RE: holomorphic binary operators over naturals; generalized hyper operators - by tommy1729 - 07/19/2012, 11:03 PM RE: holomorphic binary operators over naturals; generalized hyper operators - by JmsNxn - 07/19/2012, 11:20 PM RE: holomorphic binary operators over naturals; generalized hyper operators - by JmsNxn - 07/20/2012, 09:49 PM RE: holomorphic binary operators over naturals; generalized hyper operators - by tommy1729 - 07/21/2012, 03:42 PM RE: holomorphic binary operators over naturals; generalized hyper operators - by JmsNxn - 07/24/2012, 07:14 PM RE: holomorphic binary operators over naturals; generalized hyper operators - by JmsNxn - 08/03/2012, 06:43 PM RE: holomorphic binary operators over naturals; generalized hyper operators - by tommy1729 - 08/06/2012, 03:32 PM RE: holomorphic binary operators over naturals; generalized hyper operators - by JmsNxn - 08/08/2012, 11:23 AM RE: holomorphic binary operators over naturals; generalized hyper operators - by Gottfried - 08/09/2012, 08:59 PM RE: holomorphic binary operators over naturals; generalized hyper operators - by JmsNxn - 08/10/2012, 10:57 PM RE: holomorphic binary operators over naturals; generalized hyper operators - by Xorter - 08/18/2016, 04:40 PM RE: holomorphic binary operators over naturals; generalized hyper operators - by JmsNxn - 08/22/2016, 12:19 AM

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