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 holomorphic binary operators over naturals; generalized hyper operators JmsNxn Long Time Fellow Posts: 571 Threads: 95 Joined: Dec 2010 08/03/2012, 06:43 PM (This post was last modified: 08/04/2012, 03:13 AM by JmsNxn.) I've been experimenting looking at these operators and I've become much more familiar with its structure. I can prove convergence now. Let's suppose by contradiction that; for sufficiently large n: $\frac{x\,\,\bigtriangleup_n\,\,y}{(n\,\,\bigtriangleup_n\,\,n)^{n-s}}\,\, > \,\frac{1}{n^2}$ However. It is clear that for some $n\,\, >\,\, x,y$ 1: $n\,\,\bigtriangleup_n\,\,n\,\, >\,\, x\,\,\bigtriangleup_n\,\,y$ Therefore we write: $(n\,\,\bigtriangleup_n\,\,n)^{n-s}\,\, <\,\, n^2(x\,\,\bigtriangleup_n\,\,y)$ Now we know that $n^2\,\, <\,\, x\,\,\bigtriangleup_n\,\,y$ Therefore: $(n\,\,\bigtriangleup_n\,\,n)^{n-s}\,\, <\,\, (x\,\,\bigtriangleup_n\,\,y)^2$ However this is a contradiction because for sufficiently large $n$ the left equation becomes much larger than the right. This is easy to deduce by the relation 1 above. Therefore to prove convergence we just need add the claim: $\ln(x\,\,\bigtriangleup_n\,\,y)\,\, < \,\,x\,\,\bigtriangleup_n\,\,y$ $|\vartheta_n(s)| \,\,\le \,\,|(n\,\,\bigtriangleup_n\,\,n)^{s-n}|$ YES! We have convergence for all s. $x\,\,\bigtriangleup_s\,\,y = \prod_{n=0}^{\infty} (x\,\,\bigtriangleup_n\,\,y)^{\vartheta_n(s)}$ $\vartheta_n(s) = \frac{\sin(\pi(s-n))}{\pi(s-n)}(n\,\,\bigtriangleup_n\,\,n)^{s-n} \psi_n(s)$ An important theorem I have to prove is the following, I consider it a stern requirement of hyperoperators. For all $\Re(s) \ge 0$ and $\epsilon \,\,>\,\, 0$ and $x,y\,\,> 1$ $|x\,\,\bigtriangleup_{s+\epsilon}\,\,y| > |x\,\,\bigtriangleup_{s}\,\,y|$ I'll mull over that for awhile. also; hopefully: $\frac{d^n}{ds^n} x \,\,\bigtriangleup_s\,\,y\,\,\,\,> 0$ for at least $\Re(s) > 0$ « 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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