Till now we always discussed right-bracketed tetration, i.e. with the mother law:

a[n+1](b+1)=a[n](a[n+1]b)

here however I will introduce that balanced mother law:

a[n+1](2b) = (a[n+1]b) [n] (a[n+1]b)

a major difference to the right-bracketing hyperopsequence is that we can only derive values of the form 2^n for the right operand. Though this looks like a disadvantage, it has the major advantage being able to uniquely (or at least canonicly) being extended to the real/complex numbers.

First indeed we notice, that if we set a[1]b=a+b and if we set the starting condition a[n+1]1=a then the first three operations are indeed again addition multiplication and exponentiation:

by induction

But now the major advantage, the extension to the real numbers. We can easily see that

for

for example , , . There and so , and .

Now the good thing about each is that it has the fixed point 1 () and we can do regular iteration there. For k>2, it seems .

Back to the operation we have

or in other words we define

.

I didnt explicate it yet, but this yields quite sure and also on the positive reals.

I will see to provide some graphs of x[4]y in the future.

The increase rate of balanced tetration should be between the one left-bracketed tetration and right-bracketed/normal tetration.

I also didnt think about zeration in the context of the balanced mother law. We have (a+1)[0](a+1)=a+2 which changes to a[0]a=a+1 by substituting a+1=a. However this seems to contradict (a+2)[0](a+2)=a+4. So maybe there is no zeration here.

a[n+1](b+1)=a[n](a[n+1]b)

here however I will introduce that balanced mother law:

a[n+1](2b) = (a[n+1]b) [n] (a[n+1]b)

a major difference to the right-bracketing hyperopsequence is that we can only derive values of the form 2^n for the right operand. Though this looks like a disadvantage, it has the major advantage being able to uniquely (or at least canonicly) being extended to the real/complex numbers.

First indeed we notice, that if we set a[1]b=a+b and if we set the starting condition a[n+1]1=a then the first three operations are indeed again addition multiplication and exponentiation:

by induction

But now the major advantage, the extension to the real numbers. We can easily see that

for

for example , , . There and so , and .

Now the good thing about each is that it has the fixed point 1 () and we can do regular iteration there. For k>2, it seems .

Back to the operation we have

or in other words we define

.

I didnt explicate it yet, but this yields quite sure and also on the positive reals.

I will see to provide some graphs of x[4]y in the future.

The increase rate of balanced tetration should be between the one left-bracketed tetration and right-bracketed/normal tetration.

I also didnt think about zeration in the context of the balanced mother law. We have (a+1)[0](a+1)=a+2 which changes to a[0]a=a+1 by substituting a+1=a. However this seems to contradict (a+2)[0](a+2)=a+4. So maybe there is no zeration here.