Interesting commutative hyperoperators ?

[tommy1729]

Consider the following post made by my follower, who recycled some of my ideas :

https://math.stackexchange.com/questions...eroperator

In case that link dies or the topic gets closed I copy the text :

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After reading about Ackermann functions , tetration and similar, I considered the commutative following hyperoperator ? 

\[ F(0,a,b) = a + b \]
\[ F(n,c,0) = F(n,0,c) = c \]
\[ F(n,a,b) = F(n-1,F(n,a-1,b),F(n,a,b-1)) \]

I have not seen this one before in any official papers.
Why is this not considered ?
Does it grow to slow ? Or to fast ?

It seems faster than Ackermann or am I wrong ?

Even faster is The similar 

\[ T(0,a,b) = a + b \]
\[ T(n,c,0) = T(n,0,c) = n + c \]
\[ T(n,a,b) = T(n-1,T(n,a-1,b),T(n,a,b-1)) \]

which I got from a friend.

Notice if \(nab = 0 \) then \(T(n,a,b) = n + a + b \).

One possible idea to extend these 2 functions to real values , is to extend those “ zero rules “ to negative ones. 

So for instance for the case \(F\) :

\[ F(- n,a,b) = a + b \]
\[ F(n,-a,b) = -a + b \]
\[ F(n,a,-b) = a - b \]

The downside is this is not analytic in \(n\).


Any references or suggestions ??

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What do you guys think ?

Regards

Tommy1729

Btw im thinking about extending fake function theory to include negative numbers too, but without singularities( still entire ).