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 Interesting commutative hyperoperators ? tommy1729 Ultimate Fellow Posts: 1,493 Threads: 356 Joined: Feb 2009 02/17/2020, 11:07 PM 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 : —- 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 ?? ———- 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 ). « Next Oldest | Next Newest »

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