Rational operators (a {t} b); a,b > e solved

[JmsNxn]

what is the idea or intention behind working with base eta ?

First I noticed that at base 2 humps would appear at around 0.5 and 1.5, and if we increased the base, the humps would get sharper and sharper and taller and taller. So I figured, there must be a base value where the humps disappear altogether. My first guess was eta, since everyone talks about how closely related it is to tetration, but when I made a graph of \( f(t) = \eta \bigtriangleup_t 2 \) there were still visible humps at 0.5 and 1.5. Disappointed, and intrigued by the cheta function, I tried using the upper superfunction of eta, and voila, the values converged miraculously.

So it was more just a shot in the dark as opposed to a real mathematical deduction as to why we used base eta.

I wonder what it means that fatb(3,-1,4)=5.429897..?

no, it isn't correct, sorry.
fatb(2,-1,2) = 2º2=4 -> 2+2=4

fatb(3,-1,3) = 3º3=5 -> 3+2=5

fatb(4,-1,3) = 4º3=5 - it's correct. because that's here down:
aºb
if a>b then a+1
if b>a then b+1
if a=b then a+2 or b+2

no, this isn't zeration, this a different operator designed to preserve the ring structure of operators [t-1] and [t].

Well not on the whole complex plane, but on the real axis, wouldnt that be nice?
I anyway wonder whether thats possible at all.
As \( h_b \) is not even differentiable at 1, I wonder whether f is. Did you compare the derivations from left and right?

Yes it would be nice to have it analytic on the real axis, I think it should be potentially analytic for at least (-oo, 1). I'm just not sure about the convergence radius.

No I haven't compared the derivations, I'll do that though. It's hard for me to think up tests I can do with only highschool math under my belt

This is also in sync with the convention for quasigroups, i.e. groups with non-associative operation but with left- and right-inverse. There the left- and right inverse are written as / and \.

hmm, I really understand where you're coming from with using a standard notation instead of the triangles, it saves a lot of confusion, but I think that for the same reason the gamma function is written \( \Gamma(n+1) \) instead of \( n! \), logarithmic semi-operators should be written using their own notation--just to be clear this is only one extension of hyper operators. I know that simply stating this is the natural extension of hyper operators will step on a lot of people's toes.



Also, I have something very interesting to report!

My conjecture \( \text{cheta}(t) = e\, \bigtriangleup_t \,e \) can be proved for \( t \in (-\infty, 2], t \in Z \).

\( \text{cheta}(0) = 2e\\
\text{cheta}(-1) = \log_\eta(2e) = \log_\eta(2) + e \) and if anyone knows their deltation, like I just explained
\( a + \log_\eta(2) = a \,\,\bigtriangleup_{-1}\,\, a \)

therefore:
\( \text{cheta}(-1) = e\,\, \bigtriangleup_{-1}\,\, e \)

The proof can actually be made even simpler,
consider, \( \R(t),\R(p) \le 1; p,t \in C \):
\( log_\eta^{\circ -p}(a\,\, \bigtriangleup_t\,\, b) = log_\eta^{\circ -p}(a)\,\, \bigtriangleup_{t-p}\,\, \log_\eta^{\circ -p}(b) \)

therefore since:
\( \text{cheta}(0) = e\,\, \bigtriangleup_0\,\, e \),
\( \log_\eta^{\circ -p}\text{cheta}(0) = \log_\eta^{\circ -p}(e\,\, \bigtriangleup_0\,\, e) \)
\( \text{cheta}(p) = \log_\eta^{\circ -p}(e)\,\, \bigtriangleup_p \,\,\log_\eta^{\circ -p}(e) \), and since there's a fixpoint at e,
\( \text{cheta}(p) = e\,\, \bigtriangleup_p\,\, e \)

This proves a beautiful connection between logarithmic semi operators and the cheta function.

And also gives me a new beautiful identity:
\( e^{\large{\frac{e\,\, \bigtriangleup_{t-1}\,\, e}{e}}} = e\, \bigtriangleup_t\, e \)