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
=(a[2]2^n)+(a[2]2^n)=a2^n+a2^n=2a2^n=a2^{n+1})
=(a[3]2^n)(a[3]2^n)=a^{2^n}a^{2^n}=a^{22^n}=a^{2^{n+1}})
But now the major advantage, the extension to the real numbers. We can easily see that
)
for =x[k]x)
for example
, 
, [3](x^x)=(x^x)^{x^x}=x^{xx^x})
. There =x^x)
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 =1)
.
Back to the operation we have
)
or in other words we define
}(x))
.
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 example
Now the good thing about each
Back to the operation we have
I didnt explicate it yet, but this yields quite sure
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.
