Generalized recursive operators

[bo198214]

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.

[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 \( n\ge 2 \). Which is somehow strange to have it first a decreasing function afterwards being constant.

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).