Superfunction is a multivalued function defined over a set of functions not over a set of numbers:

means that takes a function and gives a function calles superfunction of

such that satisfies

1)

since there are infinite solution for (infinite superfunctions) that means that is multivalued and then is not a function at all and we have to put some restrictions:

using Trapmann-Kouznetsov terminology used in their paper "5+ methods..." we call the -based superfunction of the function that satifies two requirements

1)

2)

and we have

In this way we obtain uniqueness over the naturals: in fact superfunction is equivalent to the "definition by recursion" that is unique .

But is not over the reals... there we need more requirments.

Obviously this is still not enough to achieve the uniqueness of (iteration of ) that would mean having to be a function over a set of functions (not multivalued).

By the way I guess that Trapmann and Kouznetsov tried to find such additionals requirments but my math level is not enough to understand it.

Anyways we have that is a function and is the half superfunction.

example :

let define and we have

(multiplication is the 0-based superfunction of addition)

so we search for a such that

and that if we should have (maybe...)

and

I apologize if I did some mistakes.

means that takes a function and gives a function calles superfunction of

such that satisfies

1)

since there are infinite solution for (infinite superfunctions) that means that is multivalued and then is not a function at all and we have to put some restrictions:

using Trapmann-Kouznetsov terminology used in their paper "5+ methods..." we call the -based superfunction of the function that satifies two requirements

1)

2)

and we have

In this way we obtain uniqueness over the naturals: in fact superfunction is equivalent to the "definition by recursion" that is unique .

But is not over the reals... there we need more requirments.

Obviously this is still not enough to achieve the uniqueness of (iteration of ) that would mean having to be a function over a set of functions (not multivalued).

By the way I guess that Trapmann and Kouznetsov tried to find such additionals requirments but my math level is not enough to understand it.

Anyways we have that is a function and is the half superfunction.

example :

let define and we have

(multiplication is the 0-based superfunction of addition)

so we search for a such that

and that if we should have (maybe...)

and

I apologize if I did some mistakes.

MathStackExchange account:MphLee