02/24/2009, 09:57 PM
sheldonison Wrote:\( \lim_{b \to \eta^+}\text{ } \lim_{n \to \infty}
\text{sexp}_e(x) = \text{ln(ln(ln(}\cdots
\text{sexp}_b (x + \text{slog}_b(\text{sexp}_e(n)))))) \)
A very appealing formula (though the two limits belong to the right side of the equal sign)!
For my taste, if I rewrite and generalize your formula a bit ad hoc (dont know whether it is true in that form):
\( {\exp_b}^{\circ t}(x)=\lim_{a\downarrow \eta} \lim_{n\to\infty} {\log_b}^{\circ n}(\text{sexp}_a (t+\text{slog}_a({\exp_b}^{\circ n}(x)))) \)
has a big similarity to Zdun's [1] formula for regular iteration (where \( f \) is assumed to have fixed point at 0):
\( f^{\circ t}(x) = \lim_{n\to\infty} f^{\circ -n}(q^t f^{\circ n}(x)) \), \( q=f'(0) \).
Only \( q^t y \) is replaced by \( \text{sexp}_a(t+\text{slog}_a(y)) \). But be aware that \( q^ty = \exp(\log(q)t+\log(y)) \), which makes the similarity even more striking.
[1] Zdun, Marek Cezary. Regular fractional iterations. Aequationes Math. 28 (1985), no. 1-2, 73--79. MR0781210
