I wonder if hyper-operations after multiplication can be approximated by:

for and

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 in Knuth notation. I have no idea what this would be, but this function gives somewhere around 256 thousand.

Andrew Robbins

for and

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 in Knuth notation. I have no idea what this would be, but this function gives somewhere around 256 thousand.

Andrew Robbins