Could be tetration if this integral converges JmsNxn Ultimate Fellow     Posts: 1,214 Threads: 126 Joined: Dec 2010 05/05/2014, 04:27 PM (This post was last modified: 05/05/2014, 05:31 PM by JmsNxn.) (05/05/2014, 02:11 AM)mike3 Wrote: (05/04/2014, 09:06 PM)JmsNxn Wrote: $F(n) = \sum_{j=0}^n \frac{n!(-\lambda)^{n-j}}{j!(n-j)!(^j e)}$ So is this $F$ supposed to approximate tetration if $\lambda$ is small? As if so, then it doesn't seem to be working for me. If I take $\lambda = 0.01$ and the integral upper bound at 2000, I get $F(1.5)$ as ~443444.33873479713260158296678612894384. Clearly, that can't be right -- it should be between $e$ and $e^e$ (if this is supposed to reproduce the Kneser tetrational then it should be ~5.1880309584291901006085359610758671512). It gets worse the smaller you make $\lambda$ -- i.e. it doesn't seem to converge. Also, picking values to put in that are near-natural numbers doesn't seem to work, either.I'll note firstly that $F(n) = \sum_{j=0}^n \frac{n!\lambda^{n-j}}{j!(n-j)!(^j e)}$ I accidentally added an extra negative. But that doesn't really affect convergence. I understand whats happening. Hmm. That makes sense now that I think about it. I was hoping you could take lambda small but not too small and it wouldn't diverge too fast but because obviously $\lambda = 0$ diverges this doesn't happen. Maybe if you try $\lambda = 1$ I've done more research into this form of the operator so perhaps we can work with this one. $F(n) = \sum_{j=0}^n \frac{(-1)^{n-j} n!}{j!(n-j)!(^j e)}$ I do have a nice result that will work for these operators that is slightly off topic but is related to continuum sums. If $\beta = \sum_{n=0}^\infty \frac{x^n}{n!(^n e)}$ is as before and $z \in \mathbb{C}$: $\frac{d^{-z}}{dx^{-z}}|_{x=0} e^{x}\beta(-x)= F(-z) = \frac{1}{\G(z)}(\sum_{n=0}^\infty F(n) \frac{1}{n!(n+z)} + \int_1^\infty e^{-x}\beta(x)x^{z-1}\,dx)$ Quite fantabulously if $\bigtriangledown F(-z) = F(1-z) - F(-z)$ we get the really interesting result: $\bigtriangledown^n F(-z) = \frac{d^{-z}}{dx^{-z}}|_{x=0} e^x \beta^{(n)}(x)$ And of course: $[\bigtriangledown^n F(-z)]_{z=0} = \frac{1}{(^n e)}$ I'm a little fuzzy on the following but if we use a fractional iteration of $\bigtriangledown$ we can find a very nice interpolant of $(^n e)$. Since it's all using fractional calculus I have an impressionistic vision of how a similar proof of recursion would go (picking a certain fractional iteration of $\bigtriangledown$ I have in my mind and then using a similar contour integral technique). I'm really intrigued by this idea but I think I have a more general result we need. I wonder if there exists a theorem in complex analysis on the following. If $\phi(z)$ is holomorphic on $\Re(z) > -b$ does there exist some holomorphic function $\pi(z)\neq 0$ holomorphic on $\Re(z) > -b$ such that: $|\pi(z)\phi(z)|, |\pi(z)\phi(z+1)| < C e^{\alpha |\Im(z)| + \rho|\Re(z)|}$ for $0 \le \alpha < \pi/2$ and $\rho \ge 0$ such that $\pi$ satisfies some conditions I'm not sure of yet. It cannot interpolate $\phi(n)$, it cannot interpolate the inverse and it cannot be a fair amount of obvious easy functions. « Next Oldest | Next Newest »

 Messages In This Thread Could be tetration if this integral converges - by JmsNxn - 04/03/2014, 02:14 PM RE: Could be tetration if this integral converges - by sheldonison - 04/30/2014, 11:17 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/02/2014, 03:33 PM RE: Could be tetration if this integral converges - by tommy1729 - 04/30/2014, 12:29 PM RE: Could be tetration if this integral converges - by mike3 - 05/03/2014, 01:19 AM RE: Could be tetration if this integral converges - by tommy1729 - 05/11/2014, 04:26 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/11/2014, 04:30 PM RE: Could be tetration if this integral converges - by tommy1729 - 05/11/2014, 04:52 PM RE: Could be tetration if this integral converges - by mike3 - 05/03/2014, 05:24 AM RE: Could be tetration if this integral converges - by mike3 - 05/03/2014, 07:13 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/03/2014, 06:12 PM RE: Could be tetration if this integral converges - by mike3 - 05/04/2014, 02:18 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/04/2014, 07:27 PM RE: Could be tetration if this integral converges - by mike3 - 05/05/2014, 12:55 AM RE: Could be tetration if this integral converges - by mike3 - 05/04/2014, 11:50 AM RE: Could be tetration if this integral converges - by sheldonison - 05/04/2014, 03:28 PM RE: Could be tetration if this integral converges - by mike3 - 05/05/2014, 01:00 AM RE: Could be tetration if this integral converges - by sheldonison - 05/05/2014, 03:49 PM RE: Could be tetration if this integral converges - by tommy1729 - 05/04/2014, 01:25 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/04/2014, 07:36 PM RE: Could be tetration if this integral converges - by MphLee - 05/04/2014, 07:44 PM RE: Could be tetration if this integral converges - by mike3 - 05/04/2014, 10:42 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/04/2014, 11:32 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/04/2014, 09:06 PM RE: Could be tetration if this integral converges - by mike3 - 05/05/2014, 02:11 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/05/2014, 04:27 PM RE: Could be tetration if this integral converges - by mike3 - 05/05/2014, 11:45 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/06/2014, 12:11 AM RE: Could be tetration if this integral converges - by mike3 - 05/06/2014, 06:50 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/06/2014, 03:54 PM RE: Could be tetration if this integral converges - by mike3 - 05/07/2014, 03:25 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/07/2014, 03:18 PM RE: Could be tetration if this integral converges - by mike3 - 05/11/2014, 07:47 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/11/2014, 04:29 PM RE: Could be tetration if this integral converges - by mike3 - 05/11/2014, 11:26 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/12/2014, 01:44 AM RE: Could be tetration if this integral converges - by mike3 - 05/12/2014, 02:15 AM RE: Could be tetration if this integral converges - by JmsNxn - 05/12/2014, 03:32 PM RE: Could be tetration if this integral converges - by tommy1729 - 05/12/2014, 11:26 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/13/2014, 01:58 PM RE: Could be tetration if this integral converges - by JmsNxn - 05/05/2014, 06:18 PM RE: Could be tetration if this integral converges - by tommy1729 - 05/05/2014, 09:09 PM

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