Holomorphic semi operators, using the beta method

[tommy1729]

Ok I want to talk about the connection between superfunction operator , left-distributive and analytic continuation.

First the superfunction is not unique but computing what a function is the superfunction of , is almost unique ; just a single parameter usually.

If we have a function f(x,y) that is analytic in x and y and we take the superfunction F(z,x,y) (with the same method) WITH respect to x , for every fixed y , where z is the number of iterations of f(x,y) ( with respect to x ) then F(z,x,y) is usually analytic in both x and y !

Therefore the superfunction operator is an analytic operator.

this makes going from x <s> y to x <s+1> y - for sufficiently large s - preserve analyticity.

Secondly we want

x <s> 1 = x for all s.

by doing that we set going from x <s> y to x <s+1> y as a superfunction operator.

This gives us an opportunity to get analytic hyperoperators.

Combining x <s> 1 = x , the superfunction method going from x <s> y to x <s+1> y and the left distributive property to go from going from x <s> y to x <s-1> y we then get a nice structure for hyperoperators that connects to the ideas of iterations and superfunctions.

You see we then get that x <s> y is EXACTLY the y th iterate of x < s-1> y with respect to x and starting value y. If we set y = 1 then x <s> 1 = x thereby proving that it is indeed taking superfunctions *we start with x * (for all s).

This implies that

x <0> y = x + y is WRONG.

We get by the above :

x < 0 > y = x + y - 1

( x <0> 1 = x !! )

x < 1 > y = x y

( the super of +x + 1 - 1 aka +x y times )

x < 2 > y = x^y

( the super of x y ;  taking x * ... y times )

x < 3 > y = x^^ y 

( starting at x and repeating x^... )

This also allows us to compute x < n > y for any n , even negative.

That is a sketch of my idea.


Not sure how this relates to 2 < s > 2 = 4 ...

Now we only need to understand x < s > y for s between 0 and 1 but analytic at 0 and 1.

Gotta run.



Regards

tommy1729

Tom Marcel Raes

However this idea has issues as well. And with " as well " I mean to express my skepticism against all known hyperoperators.

A typical issue in fact :

let <-s> denote negative hyperoperators 

from the above 3 conditions above we get 

x <-s> y = y + 1

well at least for integer s > 0.

This is ofcourse problematic considering the superfunction operator.
And the loss of info for the x parameter.

We cannot simply say without " shame " that the *super* of x <-2> y = x <-1> y = y + 1 and the next super is not that fixed point function but x + y - 1.

It is also weird to think about functions x <-1.5> y between y + 1 and well y +1 ?!

Almost every hyperoperator proposed for generalization has similar issues :

for negative orders the equations are undefined , inconsistant ( log (0) or oo or inconsistant ( nonunique ) values  ) or we end up with identity or successor functions.

For me that is not a small issue.

The problem runs deep , i mean for almost every linear function occurence this issue arises.

And to fix the issue we could try stuff like x <1> y = x^2 + y^2 + 1 but this is not really what we want is it ??

 We could also try stuff like x<s>y = a* ( x <s-1> ( x<s>(y-1))  ) + b * ( ( x<s>(y-1)) <s-1> x ) with a + b = 1.

But that also does not seem what we want , lacks nice solutions and has similar problems ...

So we are not stuck but super stuck. 

(ok that is a bit of a joke )

 
Regards 

tomm1729