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Generalized recursive operators
#2
Whiteknox Wrote:If the operators +, *, ^, ^^, can be considered consecutive values of a sequence (with + as 1, I guess), is it possible to construct a generalized recursive function where R(1) is +, R(2) is *, and so on? In this case, can you find R(x) where x is fractional or real? I.e., is there an operator in between + and *?
Thats an interesting question.

Quote:[I am assuming that R(x) can be defined for positive integer x as follows: (where a and b are integers)
1. a R(x+1) (b+1) = a R(x) (a R(x+1) b)
2. a R(x+1) 1 = a
]

Here is however a difficulty: a R(x) 1 != a for x=1. So if we want f(x):= a R(x) 1 being a continuous function then f(1)=a+1, f(2)=a, f(3)=a, f(n)=a for . Which is somehow strange to have it first a decreasing function afterwards being constant.

Quote:Additionally, has anyone considered investigating the properties as x->inf of R(x)? I would imagine that 2 R(inf) 2 is still 4 as well as x R(inf) 1 = x.

Yeah, you can show that 2 R(n) 2 = 4 by induction. So the limit should be the same.

And a warm welcome to the forum (though I am nearly absent and just found some time to read and write).
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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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