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 holomorphic binary operators over naturals; generalized hyper operators JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 08/10/2012, 10:57 PM (This post was last modified: 08/11/2012, 09:38 PM by JmsNxn.) Right now I'm writing out some assumptions we have to put away. For example: $2 \,\,\bigtriangleup_s\,\,2 = 4 \,\,\Rightarrow\,\,2\,\,\bigtriangleup_{s+1}\,\,1 = 2$ This implies that $2\,\,\bigtriangleup_s\,\,1$ is not analytic because it is 2 for all s with real part greater than or equal to 1 and 3 when s is 0. Another one is that any continuous segment of operators is commutative or associative all the operators have to be. As well; operators in a continuous segment cannot have the same identity. The functional requirement is the following: $\vartheta_n(s+1) = \sum_{k=0}^{\infty} \vartheta_n(\mu_k) \vartheta_k(s)$ where we have: $\Pi(s) = x\,\,\bigtriangleup_s\,\,y$ $\Pi(\mu_k) = x\,\,\bigtriangleup_k\,\,\ell$ $\ell = x\,\,\bigtriangleup_{s+1}\,\,(y-1)\,\,\in\,\mathbb{N}$ And $\vartheta$ is as before. I can obtain $\Pi^{-1}(s)$ as a taylor series using lagrange inversion. So all of these functions are theoretically computable besides $\psi_n$ which is in $\vartheta_n$. So the requirement is restricted to it. I'm writing this all out trying to solve for the taylor series coefficients of $\psi$. $\vartheta_n$ is entire if $\psi_n$ is entire; so I hope it is. Thanks for the encouragement Gottfried. Like all math; it's slow progress. Little breakthroughs from time to time. « 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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