How to interpret this class of numbers?

2012-06-23T07:52:01Z

Let's say $ f(p) $ is a number defined as shown below:

$ \hspace {10 mm}f(p) = {\sqrt {2} ^ {{\sqrt {2} ^ \sqrt {2}} ^ {... \hspace {1 mm} p\hspace {1 mm} times } }} $

What I understand is:
We normally assume, presume or impose a restriction on $p$ that $p$ should be a positive integer.

My question is:
Can we allow $p$ to be any number in general?

      e.g. Can $p$ be irrational?

If $yes$, how to interpret such a number?
If $no$, why?

Thanks.

Answer by Harun Šiljak for How to interpret this class of numbers?

2012-06-25T12:51:43Z

As suggested in meta, I'll turn my comment into an answer (CW).

At this time there is no commonly accepted solution to the general problem of extending tetration to the real or complex values of n. Various approaches are mentioned in http://en.wikipedia.org/wiki/Tetration#Extension_to_real_heights .

Answer by Gottfried Helms for How to interpret this class of numbers?

2012-07-28T09:57:50Z

In addition to the answer of Harun Siljak one should perhaps mention that for the slightly different expression $$ \operatorname{dxp}\small([topexponent],[base],[iterationheight])=\operatorname{dxp}(x,t,h) $$ where $$ \operatorname{dxp}(x,t,0)= x
\\ \operatorname{dxp}(x,t,1)= t^x - 1
\\ \operatorname{dxp}(x,t,2)= t^{t^x - 1 } - 1
\\ \cdots $$ there is a solution for real $h$ based on the power series for the exponential function minus the constant term. Usually this is discussed for the function $$ \operatorname{dxp}(x,e,1)= \exp(x) - 1 $$ and fractional or even irrational heights $h$ and a parametrization for the coefficients for $$ \operatorname{dxp}(x,t,1)= \sum_{k=1}^\infty u^k{x^k \over k!}
\\ \operatorname{dxp}(x,t,h)= \sum_{k=1}^\infty \mathcal{P}(u,h,k){x^k \over k!}
\\ \text{where I wrote }u \text{ for } \ln(t) $$ where $\mathcal{P}$ denotes a polynomial in iteration-height, $\ln(t)$ and the series-index $k$.

For series like this and its iterations it is accepted, that the indicated family of iteration heights form a semigroup, where the height-parameter $h$ can be non-integer and can even be complex. This can already be found in L.Comtet's "advanced combinatorics" but also elsewhere.

Unfortunately, although the iterations of dxp() and exp() can be converted into each other (simply by a change of base) for integer heights, this is not uniquely determined for the fractional heights (the reason is, that for the same base in $b^x$ we have multiple bases $t$ in $t^x-1$ and the various $t$ give different results for the same $x$ and height $h$ if $h$ is fractional). Which then leads to the comment in the other answer, that there is not (yet) a commonly accepted interpretation for the noninteger heights in your original problem.

How to interpret this class of numbers? - MathOverflow

How to interpret this class of numbers?

Answer by Harun Šiljak for How to interpret this class of numbers?

Answer by Gottfried Helms for How to interpret this class of numbers?