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 where then under certain conditions

Now of course, the problem is that when 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 is a sequence of complex numbers such that is entire. Then, there always exists such that, is entire and Weyl differintegrable on all of and

is such that exists for all z.

If this theorem is shown, then... define

and

and we are done.

Any one have any advice on how I can show this theorem? this is quite a stump.

Well I'll start by saying, if where then under certain conditions

Now of course, the problem is that when 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 is a sequence of complex numbers such that is entire. Then, there always exists such that, is entire and Weyl differintegrable on all of and

is such that exists for all z.

If this theorem is shown, then... define

and

and we are done.

Any one have any advice on how I can show this theorem? this is quite a stump.