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 Theorem in fractional calculus needed for hyperoperators JmsNxn Long Time Fellow Posts: 464 Threads: 85 Joined: Dec 2010 07/03/2014, 02:17 PM Hey everybody! Well I've boiled down my requirements for solving tetration, pentation, semi operators, and a whole list of recursive relationships using fractional calculus into a single theorem. I am pretty certain this theorem will be true. Well I'll start by saying, if $f(x) = \sum_{n=0}^\infty a_n x^n/n!$ where $F(a_n) = a_{n+1}$ then under certain conditions $F(\frac{d^{z}}{dx^z}|_{x=0}f(x)) = \frac{d^{z+1}}{dx^{z+1}}|_{x=0}f(x)$ Now of course, the problem is that when $a_n$ is something like tetration, or pentation, or whatever, this doesn't converge and we are stuck in the mud. So I've boiled a way to fix this. Now I don't have this theorem yet, but if its solved, all that's required is a bunch of lemmas I know how to prove and we will have tetration, pentation, hexation, semi operators, and some more. So without further ado, here is the theorem we need. Assume $a_n$ is a sequence of complex numbers such that $f(x) = \sum_{n=0}^\infty a_n \frac{x^n}{n!}$ is entire. Then, there always exists $b_n$ such that, $g(x) = \sum_{n=0}^\infty b_n \frac{x^n}{n!}$ is entire and Weyl differintegrable on all of $\mathbb{C}$ and $h(x) = \sum_{n=0}^\infty a_n b_n \frac{x^n}{n!}$ is such that $\frac{d^z}{dx^z}{|}_{x=0} h(x)$ exists for all z. If this theorem is shown, then... define $G(z) = \frac{\frac{d^z}{dx^z}|_{x=0} h(x)}{\frac{d^z}{dx^z}|_{x=0} g(x)}$ and $F(G(z)) = G(z+1)$ and we are done. Any one have any advice on how I can show this theorem? this is quite a stump. « Next Oldest | Next Newest »

 Messages In This Thread Theorem in fractional calculus needed for hyperoperators - by JmsNxn - 07/03/2014, 02:17 PM RE: Theorem in fractional calculus needed for hyperoperators - by MphLee - 07/03/2014, 04:23 PM RE: Theorem in fractional calculus needed for hyperoperators - by JmsNxn - 07/03/2014, 04:52 PM RE: Theorem in fractional calculus needed for hyperoperators - by MphLee - 07/03/2014, 05:29 PM RE: Theorem in fractional calculus needed for hyperoperators - by JmsNxn - 07/03/2014, 06:13 PM RE: Theorem in fractional calculus needed for hyperoperators - by MphLee - 07/07/2014, 06:47 PM

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