Superroots and a generalization for the Lambert-W

[tommy1729]

The thing is solving (x_m ^ x_m)^[m] = y is only close to solving
X_n^^[n] = y ( n = m in value )

True. But having this way a (non-trivial) vector of different exponents (or better: bases) which comes out to be a meaningful "nested exponentiation" I'm curious, whether one can do something with it, for instance weighting, averaging, or multisecting that sequence of exponents/bases when re-combining them to a "nested exponential". We have not yet many examples of "nested exponentiations" with a meaningful outcome.
For instance, the construction of the Schroeder-function is based on (ideally) infinite iteration of the base-function to get a linearization. If we iterate the h2()-function infinitely, the curve of the consecutive values in an x/y-diagram (where x is the iteration number) approach a horizontal line; don't know whether using that linearization shall prove useful for something similar.

(When Euler found his version of the gamma-function, that was in one version putting together sequences of integer numbers weighting and repeating in a meaningful way; there is some infinite product-representation for his gamma-function I think I recall correctly... )

(see also the updates in my previous (introducing) posting)

This reminds me of one of my posts on the OEIS many years ago , also under the pseudo tommy1729 ( i have other pseudo too ).

Go to Oeis and enter tommy1729.
Or use this link:

https://oeis.org/search?q=Tommy1729&lang...&go=Search

In particular

https://oeis.org/A102575

In abuse notation this becomes

(1 + 1/n)^^

And it was a special case of my investigations in the Tommy-Zeta functions between 2001 and 2009 given by

(1+1^(-s))^(1+2^(-s)) ...

This is somewhat similar and thus might intrest you.

Regards

Tommy1729

[andydude]

I think super-roots are important.

Iterated exponentials (\( w = \exp_x^{y}(z) \)) are a function of three variables (trivariate? function), and so they have 3 inverse functions: negatively iterated exponentials (solving for z), trivariate super-logarithms (solving for y), and trivariate super-roots (solving for x). Trivariate super-logarithms can be expressed with bivariate super-logarithms, and so are not fundamental operations, but trivariate super-roots have no known expression in terms of bivariate super-roots, and so are, so far, a fundamental operation so far as I know.

My recent research into super-roots have convinced me that we know more about them than we think we know. We can calculate the derivatives of them to a rational number in some cases, and to any precision in other cases. Using a combination of power series and Lagrange inverse series, we can calculate many many things about them, but we still don't have a closed form for these apparently useful functions. I think that given enough time, effort, and insight, we can find at least a recurrence equation that expresses how to find super-root (n + 1) given complete knowledge of super-root (n).

I'm going to go out on a limb and make a notation for these trivariate super-roots:

• \( \sqrt[y]{w}^{(z)}_{\mathrm{s}} = x \) iff. \( w = \exp_x^y(z) \)
  

One of the advantages of trivariate super-roots is that they have more algebraic identities regarding them:

• \( \sqrt[y]{w}^{(z)}_{\mathrm{s}}
  = \sqrt[y]{x^w}^{(x^z)}_{\mathrm{s}}
  = \sqrt[(y-1)]{w}^{(x^z)}_{\mathrm{s}}
  = x \)
  

and there is one about the third super-root:

• \( \sqrt[3]{w}^{(z)}_{\mathrm{s}}
  = \left(\sqrt[2]{w^z}^{\left(\frac{x^z}{z}\right)}_{\mathrm{s}}\right)^{1/z}
  = x \)
  

and there is one about the second super-root:

• \( \sqrt[2]{w}^{(z)}_{\mathrm{s}}
  = \left(\sqrt[2]{w^z}^{(1)}_{\mathrm{s}}\right)^{1/z}
  = x \)
  

where \( \sqrt[2]{w}^{(1)}_{\mathrm{s}} \) just means the bivariate super-root \( \sqrt[2]{w}_{\mathrm{s}} \).

If we could find a general way of expressing trivariate super-roots in terms of bivariate super-roots, then I think we would know much more about tetration than we do today. Perhaps along the way we will discover something new that will shed some light on super-logarithms, too, perhaps.

Regards,
Andrew Robbins

[andydude]

I believe I may have found a closed form for the power series of the third tetrate function as well.
I'm not sure if these are known, but I just used the elementary properties of binomials and Stirling numbers to derive these:

\(
\begin{equation}
{}^{3}x =
\sum_{k=0}^{\infty}
\log(x)^k
\sum_{j=0}^{k}
\sum_{i=0}^{k - j - 1}
\frac{(k - j - i)^j j^i}{(k - j - i)!j!i!}
\end{equation}
\)

\(
\begin{equation}
{}^{3}x =
\sum_{k=0}^{\infty}
(x - 1)^k
\sum_{j=0}^{k}
\sum_{J=0}^{j}
\sum_{i=0}^{k}
\sum_{I=0}^{i}
{\left[{i \atop I}\right]}
{\left[{j \atop J}\right]}
{\left({J \atop {k - j - i}}\right)}
\frac{J^I}{j!i!}
\end{equation}
\)

The first one (logarithmic power series) reminds me of something in one of Galidakis' papers about tetration, but I don't remember which paper. The second one is derived from the fact that the generating function of the signed Stirling numbers the first kind is \( (1 + x)^z \).