Hello forum.
So I've finally hit a dead end in my hyperoperator theory. "Dead end" as in I fear I've discovered that there is, in general, no closed for for [x]a, where [x] is my hyperoperator (with argument x) and a is the argument.
So this is what I've got so far: Amateur document
(The real meat of the theory starts in Section 4)
Crude summary
Define a binary operation,
, where
,
,
and so on. My goal is to get to a point where I can evaluate
for
. Next, define a unary operation,
. Denote iteration of the binary and unary operators via a superscript:
e.g.[x]b)[x]b)

Next, establish the following axioms:
A0:
A1:
A2:
(perhaps this is not really an axiom based on the way I have defined iteration of the unary operator)
A3:
From these "axioms" the following relations, among others, can be proven (see the document for derivations):
is not tetration (sorry forum)
[x]c=a[x](b\cdot c))

(definition of inverse)
(2 is a fixed point, explaining why
, etc.)
The problem:
I find that
, where
is the yth super-iteration of
.
What do I mean by "super-iteration" and how do I come to that conclusion? It's in Section 4.3 of the document, but here it is again anyway:
Given the axioms, we can find that
. Define
. Now we can iterate both sides
times to obtain
.
Now we must define
, and repeat the
iteration to obtain
.
Repeating this algorithm y times results in
.
I know that the "super-iteration" method is necessary because I've tried other more naive derivations of
which turned out to be wrong when testing various combinations of x and y (e.g.
should be the same for (x,y)=(0,1) and (x,y)=(1,0)).
So anyways, I'm putting this out here on the forum because super-iteration seems to be an unmanageable concept to me, and I have not been able to find a way to avoid it. I hope that someone with more experience in this area of mathematics will have some idea of how to manipulate the super-iteration into something simpler or how to relate
and
without the need for super-iteration. Ideally, the end result is a closed form expression for
, where x can be any complex number.
I'm using the term "super-iteration" because it is used on another thread on the forum (which, btw, is the only place on the internet where I could find information on anything similar to what I'm encountering).
Thanks for reading. I know that this is my problem, but if anyone finds it interesting enough to think about, I would appreciate any comments on the validity of what I have so far and suggestions on how to proceed.
Thanks again.
hixidom
So I've finally hit a dead end in my hyperoperator theory. "Dead end" as in I fear I've discovered that there is, in general, no closed for for [x]a, where [x] is my hyperoperator (with argument x) and a is the argument.
So this is what I've got so far: Amateur document
(The real meat of the theory starts in Section 4)
Crude summary
Define a binary operation,
e.g.
Next, establish the following axioms:
A0:
A1:
A2:
A3:
From these "axioms" the following relations, among others, can be proven (see the document for derivations):
The problem:
I find that
What do I mean by "super-iteration" and how do I come to that conclusion? It's in Section 4.3 of the document, but here it is again anyway:
Given the axioms, we can find that
Now we must define
Repeating this algorithm y times results in
I know that the "super-iteration" method is necessary because I've tried other more naive derivations of
So anyways, I'm putting this out here on the forum because super-iteration seems to be an unmanageable concept to me, and I have not been able to find a way to avoid it. I hope that someone with more experience in this area of mathematics will have some idea of how to manipulate the super-iteration into something simpler or how to relate
I'm using the term "super-iteration" because it is used on another thread on the forum (which, btw, is the only place on the internet where I could find information on anything similar to what I'm encountering).
Thanks for reading. I know that this is my problem, but if anyone finds it interesting enough to think about, I would appreciate any comments on the validity of what I have so far and suggestions on how to proceed.
Thanks again.
hixidom