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 ). |