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Let a>0. Does there exist an analytic function r(x) on (-2,+¥) that satisfies the following:

1) r(0)=1, and

2) r(x+1)=a^{r(x)}

for all x in (-1,+¥)???

Note 1. An approximation to this function is

s(x)=log_{a}(x+2) , for x in (-2,-1],

s(x)=x+1, for x in (-1,0],

s(x)=a^{x} , for x in (0,1],

s(x)=a^{(a(x-1))}, for x in (1,2],

s(x)=a^{(a(a(x-2)))}, for x in (2,3], etc.

Note 2. a=e^{(1/e)} is the largest value of a for which lim_{x®¥}r(x)=L for some L in R. For this case (a=e^{(1/e)} ), L=e.

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Partial answer (for partial marks :-)).

a>0. Consider the following function, for the time being acting only over the positive integers: f(n)=a^{||n||}=a^{(a(a(a...(aa))))}, n many. We extend f to the set of rationals in the "usual" way : x^{||1/m||} is the inverse of x^{||m||} over positive real numbers; that is, y= x^{||1/m||} if x= y^{||m||}, where y^{||m||} is the invertible function defined by y^{||m||}=y^{yy..yy}} m many times (here y is the variable, m is fixed). So then, f(1/m)= a^{||1/m||}; further, f(n/m)=(a^{||1/m||})^{||m||}. This gives f over rationals. Then extend to positive real numbers in the "usual" way : take a sequence r_{1}, r_{2},... of rationals converging to a real number r and define a^{r} = lim_{i®¥}r_{i}.

So we got f(x)= a^{||x||}, where x is real positive. It appears ("appears" is always a gap) that everything required is fulfilled except for the fact that f is defined over (0, ¥) rather than over (-2, ¥). I do not know if f could be extended. But the question could certainly be changed to accommodate the answer :-).

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There are some problems with such an approach.

Whereas for exponents, we have, for example that

(x^4)^(1/2)=x^2, it would not hold in this case.

If we take a=1.5, according to your definition, a^^2 (meaning a^a in your notation) is approximately 1.837117307.

a^^4 is approximately 2.349005319

finally, (a^^4)^^(1/2) is roughly 1.66839753, which is different from a^^2

In general, the rules for exponents

(a^bc)^(1/bd)=(a^c)^(1/d)=(a^(1/d))^c

do not hold for our "hyper-exponents".

Here is a second example:

(2^^4)^^(1/3)=2.58611054

However, (2^^(1/3))^^4=2.180596631

Perhaps, and this is just a conjecture, we could take a^^(p/q)=lim(n->infinity)(a^^(pn))^^(1/(qn))

if such a limit exists and equals

lim(n->infinity)(a^^(1/(qn))^^(pn).

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You are right: the extenstion to rationals is not well defined unless a^^(p/q) is equal to a^^(np/nq). And that is not true in general (essentially indicated in your last post). There is in fact one more problem: the function x^^n (fixed positive integer n) is not (necessarily) invertible over the set of positive real numbers x. For example, x^^4 looks like this:

We can surely bypass this by restricting to reals not less than 1. Under that restriction, your idea with limits seems reasonable - but, of course, needs to be worked out.