08/10/2007, 09:01 PM
Interesting you mention base-(\( e^{1/e} \)) tetration, because I also beleive that this base is the only base for which convergence can be proven, although convergence of series for other bases might in fact converge.
Also, there are essentially 3 types of series expansions of iterated exponentials (one for each argument). There is Galidakis' expansions (around the hyper-base), Geisler's expanions (around the main argument), and my expansions (around the hyper-exponent, but inverse). So naturally these are very difficult to compare, if that is indeed what you want to do.
Most general iteration expansions involve series expansions around the main argument (x), where the coefficients are functions of t (iteration/time), but the easiest way to solve functional equations usually involves a series in t whose coefficients are functions of x. So if AR_b(z) is my expansion and DG_bt(x) is Geislers extension, and IG_t(b) is Galidakis' extension, they could be compared as:
Andrew Robbins
Also, there are essentially 3 types of series expansions of iterated exponentials (one for each argument). There is Galidakis' expansions (around the hyper-base), Geisler's expanions (around the main argument), and my expansions (around the hyper-exponent, but inverse). So naturally these are very difficult to compare, if that is indeed what you want to do.
Most general iteration expansions involve series expansions around the main argument (x), where the coefficients are functions of t (iteration/time), but the easiest way to solve functional equations usually involves a series in t whose coefficients are functions of x. So if AR_b(z) is my expansion and DG_bt(x) is Geislers extension, and IG_t(b) is Galidakis' extension, they could be compared as:
- series-transpose to b of DG_bt(1) = IG_t(b)
- series-transpose to t of DG_bt(1) = series-inverse of (AR_b(z)=t) = AR^-1_b(t)
Andrew Robbins