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 ??