Interesting commutative hyperoperators ? - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Interesting commutative hyperoperators ? (/showthread.php?tid=1255) Interesting commutative hyperoperators ? - tommy1729 - 02/17/2020 Consider the following post made by my follower, who recycled some of my ideas : https://math.stackexchange.com/questions/3550548/new-commutative-hyperoperator 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 ).