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On to C^\infty--and attempts at C^\infty hyper-operations
Hey Everyone. Haven't been on for a while, been a little busy.  I thought I'd post the modified form of the paper.

It details how to construct hyper-operations; minus maybe a few details. But I'm confident I got everything down. In fact, I'm confident that if we do it in the manner Sheldon describes, using a sequence of infinite compositions,

Not much really changes.  I chose to use the exponential convergents rather than the convergents, simply because it generalizes well to the construction of arbitrary super-functions. All we need really is an exponential convergents. In fact, will look pretty much similarly, because of its exponential nature. The better form of Sheldon's method is that we retain holomorphy. Since I'm only trying to show , I figure holomorphy isn't really needed.

I do believe we could definitely use to construct . I believe it's more of an aesthetic issue, and I find it a bit more natural to just use,

And do away with . Also because this satisfies the more natural equation,

And it removes a bit of the untangling if we were to use .

Anyway, here's what I have so far. The proof of hyper-operators is surprisingly copy/paste from the proof of tetration, so long as you pay attention to the generalization, it should be fine.

Any questions, comments or the what have you are greatly appreciated. I spent a lot of time restructuring the paper, but a lot of it is similar to what I wrote before.

Thanks, James

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Messages In This Thread
RE: On to C^\infty--and attempts at C^\infty hyper-operations - by JmsNxn - 03/02/2021, 09:55 PM

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