The upper superexponential

[bo198214]

Does this upper super expoonential equation also hold for b=\( e^{1/e} \)?

Interesting question. Unfortunately the convergence gets quite bad for \( b \) approaching \( e^{1/e} \), so I could not really check numerically.
On the other hand Walker describes also two solutions for \( b=e^{1/e} \) in "On the solutions of an Abelian equation". I did not really read this article, but I think he also showed that these solutions are not the limit of approaching \( e^{1/e} \).

Is this "chi" the same as the "Chi distribution" used in probability?

No, not at all. Its just somewhat similar to "Sch" in Schroeder.

Any links to a definition for
\( \chi \) and \( \chi^{-1} \)

Ya, for example in the thread regular slog.
Literature is: Szekeres "Regular iteration of real and complex functions."