11/23/2007, 06:42 AM

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 *?

[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

]

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.

[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

]

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.