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 holomorphic binary operators over naturals; generalized hyper operators JmsNxn Long Time Fellow Posts: 566 Threads: 94 Joined: Dec 2010 07/19/2012, 10:26 PM (This post was last modified: 07/19/2012, 10:31 PM by JmsNxn.) Yes. By definition. $\vartheta_n(n) = 1$ and $\vartheta_k(n) = 0 \,\,\Leftrightarrow\,\, k \neq n\,\,k \in \mathbb{N}_0$ therefore the rest of the product disappears at natural values. That's the whole trick . I'd take the limit as $\lim_{h \to n}\vartheta_n (h) = \vartheta_n(n) = 1$ just for technicality sake. $x\,\,\bigtriangleup_0\,\, y = (x + y)^1 \cdot (x \cdot y)^0 \cdot (x^y)^0 \cdot ...$ $x \,\,\bigtriangleup_1\,\, y = (x + y)^0 \cdot (x \cdot y)^1 \cdot (x^y)^0 \cdot ...$ It's real for real $s$ and complex for complex $s$ and natural for natural $s$. Very nice. « 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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