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Generalized recursive operators
#5
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:

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

   
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Messages In This Thread
Generalized recursive operators - by Whiteknox - 11/23/2007, 06:42 AM
RE: Generalized recursive operators - by bo198214 - 11/23/2007, 08:41 AM
RE: Generalized recursive operators - by andydude - 11/25/2007, 01:02 AM
RE: Generalized recursive operators - by andydude - 11/29/2007, 04:45 AM
RE: Generalized recursive operators - by andydude - 11/29/2007, 05:55 AM
RE: Generalized recursive operators - by andydude - 11/29/2007, 06:20 AM
RE: Generalized recursive operators - by andydude - 11/30/2007, 06:12 PM
RE: Generalized recursive operators - by andydude - 11/30/2007, 09:18 PM
RE: Generalized recursive operators - by bo198214 - 03/07/2008, 06:58 PM
RE: Generalized recursive operators - by Ivars - 02/02/2008, 10:11 PM
RE: Generalized recursive operators - by Ivars - 02/03/2008, 10:41 AM
RE: Generalized recursive operators - by andydude - 02/11/2008, 09:47 PM
RE: Generalized recursive operators - by Ivars - 02/14/2008, 06:05 PM
RE: Generalized recursive operators - by GFR - 02/03/2008, 04:12 PM
RE: Generalized recursive operators - by Ivars - 02/03/2008, 08:48 PM
RE: Generalized recursive operators - by GFR - 02/06/2008, 02:44 PM
RE: Generalized recursive operators - by Ivars - 02/06/2008, 02:56 PM
RE: Generalized recursive operators - by Ivars - 02/06/2008, 03:43 PM
RE: Generalized recursive operators - by GFR - 03/10/2008, 09:53 PM
RE: Generalized recursive operators - by GFR - 03/11/2008, 10:24 AM
RE: Generalized recursive operators - by bo198214 - 03/11/2008, 10:53 AM
RE: Generalized recursive operators - by GFR - 03/12/2008, 12:13 AM
RE: Generalized recursive operators - by GFR - 03/13/2008, 06:41 PM
RE: Generalized recursive operators - by Stan - 04/04/2011, 11:52 PM

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