We all know of right hand tetration, evaluated right to left, and left hand tetration, evaluated left to right. which are both defined with different purposes in mind. Right handed tetration is the superfunction of exponentiation, left handed tetration is analytic very simply.

But what about an operator that continues the distributivity found across (exponentiation, multiplication) and (multiplication, addition).

Or in other words, can such an operator exist? if:

And if such is the case, what is it's operator generating function in the sense that:

would then be the superfunction of .

this would make our distributive tetration commutative, if of course, such a function exists.

The term "operator generating function" is novel and is generalized from the fact that exponentiation is the operator generating function of multiplication, insofar as:

and the identity function is the operator generating function of addition.

Interesting things occur with this new operator. I'll start by listing them now:

which occurs more generally as:

which means if this operator turns out to be consistent, then we have a very handy tool for base conversion.

and it also means:

which says more simply:

which says tetration spreads across the same way differentiation spreads across convolution.

We also have weird results for when 1 is an argument:

and therefore:

This gives the limit:

which is a law given by all operator generating functions.

we also find that:

which means

So therefore the only way to keep this working is to have:

Therefore 's inverse is either multivalued, or on the real axis increases from zero to one from across negative infinity to infinity. I'm hoping then that complex values in the domain open up the range a whole lot more.

the identity of is

If we define the inverse of as

and, if is the superfunction of :

then we are given:

so that:

and

which is a general law for all operators with a operator generating function.

Anyway, my question to all of you, is if any of this is consistent. I always have trouble poking holes in the ideas I have, and other people seem to be a lot sharper at doing it. Therefore I ask you guys, my fellow tetrationers.

regards, James

But what about an operator that continues the distributivity found across (exponentiation, multiplication) and (multiplication, addition).

Or in other words, can such an operator exist? if:

And if such is the case, what is it's operator generating function in the sense that:

would then be the superfunction of .

this would make our distributive tetration commutative, if of course, such a function exists.

The term "operator generating function" is novel and is generalized from the fact that exponentiation is the operator generating function of multiplication, insofar as:

and the identity function is the operator generating function of addition.

Interesting things occur with this new operator. I'll start by listing them now:

which occurs more generally as:

which means if this operator turns out to be consistent, then we have a very handy tool for base conversion.

and it also means:

which says more simply:

which says tetration spreads across the same way differentiation spreads across convolution.

We also have weird results for when 1 is an argument:

and therefore:

This gives the limit:

which is a law given by all operator generating functions.

we also find that:

which means

So therefore the only way to keep this working is to have:

Therefore 's inverse is either multivalued, or on the real axis increases from zero to one from across negative infinity to infinity. I'm hoping then that complex values in the domain open up the range a whole lot more.

the identity of is

If we define the inverse of as

and, if is the superfunction of :

then we are given:

so that:

and

which is a general law for all operators with a operator generating function.

Anyway, my question to all of you, is if any of this is consistent. I always have trouble poking holes in the ideas I have, and other people seem to be a lot sharper at doing it. Therefore I ask you guys, my fellow tetrationers.

regards, James