Just an idea:

Is it possible to derive tetration and above starting from self root concept and defining tetration as an inverse to it?

With all branches? And then see the conection to [n]-tation? Self root seems not so abstract and infinite, yet it yields inverse tetration nicely. What if me make it an starting axiom?

Of course, to include branches of Lambert function and h, will have to accept that each self root has infinity of values.Some of which might be real.

One example:

h(i^(1/i))= i

h((1/i)^(i) = -i

So we get nicely and logically first to branches of Lambert Wo, W-1. They both correspond to one real input,

i^(1/i)=(1/i)^i=e^pi/2 , but we get 2 different self roots as we rearange x^(1/x) into (1/x)^x.

The next branches should be attainable by taking self roots of i = e^i 5pi/2 instead of e^(ipi/2), so MAYBE:

h(e^(i5pi/2)^(e-i5pi/2))= +- next 2 branches of W in combination with some branches of logarithm.

Superroot might be attainable by using negative values of -i=e^(I*3pi/2), 1/-i=e^(-(I*3pi/2) I have to check.

May be this is wrong again or done 200 times with no use, please let me know.

Ivars

Moderator's note: Splitted from "Generalized recursive operations"

Is it possible to derive tetration and above starting from self root concept and defining tetration as an inverse to it?

With all branches? And then see the conection to [n]-tation? Self root seems not so abstract and infinite, yet it yields inverse tetration nicely. What if me make it an starting axiom?

Of course, to include branches of Lambert function and h, will have to accept that each self root has infinity of values.Some of which might be real.

One example:

h(i^(1/i))= i

h((1/i)^(i) = -i

So we get nicely and logically first to branches of Lambert Wo, W-1. They both correspond to one real input,

i^(1/i)=(1/i)^i=e^pi/2 , but we get 2 different self roots as we rearange x^(1/x) into (1/x)^x.

The next branches should be attainable by taking self roots of i = e^i 5pi/2 instead of e^(ipi/2), so MAYBE:

h(e^(i5pi/2)^(e-i5pi/2))= +- next 2 branches of W in combination with some branches of logarithm.

Superroot might be attainable by using negative values of -i=e^(I*3pi/2), 1/-i=e^(-(I*3pi/2) I have to check.

May be this is wrong again or done 200 times with no use, please let me know.

Ivars

Moderator's note: Splitted from "Generalized recursive operations"