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 holomorphic binary operators over naturals; generalized hyper operators JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 07/19/2012, 05:49 PM (This post was last modified: 07/19/2012, 10:19 PM by JmsNxn.) If we want. We can try to treat this as a uniqueness claim for extending hyperoperators to complex numbers. That if we find a formula for $\vartheta$ that satisfies recursion for natural numbers it should also satisfy recursion for complex numbers. Ergo; uniqueness for how hyper operators behave for complex numbers. It's not very lucid yet. Again; one step at a time. EDIT: evaluation of limit $\begin{eqnarray*} \vartheta_n(s) &=& \frac{s}{n} e^{e^s-e^n} \prod_{k=1}^{\infty}\frac{1-\frac{s}{k}}{1-\frac{n}{k}}e^{ \frac{n - s}{k}}\\ &=& \frac{s e^{e^s} \prod_{k=1}^{\infty}(1-\frac{s}{k})e^{\frac{-s}{k}}}{n e^{e^n} \prod_{k=1}^{\infty}(1-\frac{n}{k})e^{\frac{-n}{k}}} = \frac{\psi(s)}{\psi(n)}\\ \end{eqnarray*}$ $\begin{eqnarray*} f(t) &=& \ln(x\,\, \bigtriangleup_{t}\,\, y)\\ &=& \ln(x+y) \frac{\psi(t)}{t} + \sum_{k=1}^{\infty} \frac{\psi(t)}{\psi(k)} \ln(x\,\, \bigtriangleup_{k}\,\, y)\\ &=& \psi(t) \cdot (\frac{\ln(x+y)}{t} + \sum_{k=1}^{\infty} \frac{\ln(x\,\, \bigtriangleup_{k}\,\,y)}{\psi(k)} ) \end{eqnarray*}$ Therefore we can rephrase the limit as; keeping in mind the first term $\ln(x+y)/t$ disappears: $\begin{eqnarray*} L &=& \lim_{t \to \infty} | \frac{f(t+1)}{f(t)} \cdot \frac{\vartheta_{t+1}(s)}{\vartheta_{t}(s)}|\\ &=& \lim_{t \to \infty} | \frac{\psi(t+1) \vartheta_{t+1}(s) } {\psi(t) \vartheta_{t}(s)} \cdot \frac{ \frac{\ln(x+y)}{t+1} + \sum_{k=1}^{\infty} \frac{f(k)}{\psi(k)}}{\frac{\ln(x+y)}{t} + \sum_{k=1}^{\infty} \frac{f(k)}{\psi(k)} } |\\ &=&1 \end{eqnarray*}$ Which means convergence is inconclusive. Using this method generally we will have this result; so long as $\vartheta$ remains a quotient. Anyone know any other convergence tests? I'm pretty sure this outlaws the root test as inconclusive as well. The integral test? I'll keep looking... Maybe I made a mistake. I tried using the integral test and I got divergence. The methods were a little questionable however. I'll hold out on posting that just yet. « 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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