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 holomorphic binary operators over naturals; generalized hyper operators tommy1729 Ultimate Fellow Posts: 1,491 Threads: 355 Joined: Feb 2009 07/19/2012, 10:16 PM (07/19/2012, 04:44 PM)JmsNxn Wrote: We start by defining a sequence of analytic functions that obey the following rules: $\vartheta_n : \mathbb{C} \to \mathbb{C}$ $\vartheta_n(n) = 1$ and $k \in \mathbb{N}_0\,\,\,\vartheta_n(k) = 0 \,\,\Leftrightarrow\,\, k \neq n$ Necessarily; these functions can be arbitrary and they pose the true uniqueness criterion. Nonetheless we suggest these functions and give a plausible solution. ${\bf I}:\,\,\,\,\vartheta_n(s) = \frac{s}{n} e^{e^s-e^n} \prod_{k=1}^{\infty}\frac{1-\frac{s}{k}}{1-\frac{n}{k}}e^{ \frac{n - s}{k}}\,\,\,;\,\,\,n \ge 1$ $\vartheta_0(s) = e^{e^s-1} \prod_{k=1}^{\infty}(1-\frac{s}{k})e^{-\frac{s}{k}}$ Reason for this is it has fast convergence and satisfies the required conditions. Now we perform a trick. Considering hyper operators; which are written by the following: $x\,\,\bigtriangleup_{s}\,\,(x \,\,\bigtriangleup_{s+1}\,\,y) = x \,\,\bigtriangleup_{s+1}\,\,(y+1)$ $x \,\,\bigtriangleup_0\,\,y = x + y$ and the identity of $\bigtriangleup_n$ for any natural $n$ is $1$. We get the usual sequence where zero is addition, one is multiplication, two is exponentiation, etc... We then write that complex operators are products of natural operators with complex exponents. Here; $\vartheta_n$ is as before. ${\bf II}:\,\,\,\,x \,\,\bigtriangleup_s\,\, y = \prod_{n=0}^{\infty}(x\,\,\bigtriangleup_n\,\,y)^{\vartheta_n(s)} = (x + y)^{\vartheta_0(s)} \cdot (x\cdot y)^{\vartheta_1(s)} \cdot (x^y) ^{\vartheta_2(s)} \cdot (^y x)^{\vartheta_3(s)} \cdot ...$ Dont you need to prove the identity of the infinite product ? e.g. does the infinite product formula hold for x + y and x*y ?? regards tommy1729 « 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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