I wonder if hyper-operations after multiplication can be approximated by:
\( a R(n) b \approx a^b \) for \( 0\le b \le 1 \) and \( n \ge 3 \)
If so, then maybe this could be used to extend the right hyper-operations to at least a function defined over all real N, but I don't think its continuous.
Here is some Mathematica code that I used to investigate this idea:
It makes some pretty amazing graphs, but I'll have to do some more investigating before I can come up with any conclusions.
For those of you who are confused by Mathematica code, a little primer: "_" means "any", i.e. a variable, "/;" means "where", and in order to use the function you would write "Hy[3.5][E, Pi]" for example would calculate \( e {\uparrow}^{1.5} \pi \) in Knuth notation. I have no idea what this would be, but this function gives somewhere around 256 thousand.
Andrew Robbins
\( a R(n) b \approx a^b \) for \( 0\le b \le 1 \) and \( n \ge 3 \)
If so, then maybe this could be used to extend the right hyper-operations to at least a function defined over all real N, but I don't think its continuous.
Here is some Mathematica code that I used to investigate this idea:
Code:
Hy[1] := Plus;
Hy[2] := Times;
Hy[3] := Power;
Hy[n_][x_, 0] /; (n > 3) := 1;
Hy[n_][x_, 1] /; (n > 2) := x;
Hy[n_][x_, y_] /; (1 < n < 3) :=
Simplify[Evaluate[InterpolatingPolynomial[{
{1, x + y},
{2, x*y},
{3, x^y}}, n]]];
Hy[n_][x_, y_] /; (n > 3) := Piecewise[{
{HyLog[n-1][x, Hy[n][x, y + 1]], y < 0},
{x^y, 0 <= y <= 1},
{Hy[n-1][x, Hy[n][x, y - 1]], y > 1}
}];
HyLog[1] := Subtract;
HyLog[2] := Divide;
HyLog[3] := Log;
HyLog[n_][x_, 1] /; (n > 3) := 0;
HyLog[n_][x_, x_] /; (n > 2) := 1;
HyLog[n_][x_, z_] /; (1 < n < 3) :=
Simplify[Evaluate[InterpolatingPolynomial[{
{1, z - x},
{2, z/x},
{3, Log[x, z]}}, n]]];
HyLog[n_][x_, z_] /; (n > 3) := Piecewise[{
{HyLog[n][x, Hy[n-1][x, z]] - 1, z < 1},
{Log[x, z], 1 <= z <= x},
{HyLog[n][x, HyLog[n-1][x, z]] + 1, z > x}
}];It makes some pretty amazing graphs, but I'll have to do some more investigating before I can come up with any conclusions.
For those of you who are confused by Mathematica code, a little primer: "_" means "any", i.e. a variable, "/;" means "where", and in order to use the function you would write "Hy[3.5][E, Pi]" for example would calculate \( e {\uparrow}^{1.5} \pi \) in Knuth notation. I have no idea what this would be, but this function gives somewhere around 256 thousand.
Andrew Robbins