01/13/2017, 07:47 PM
I was wondering, could you show to me exactly how you're proving that your series equals tetration on the naturals? This is the only thing I don't quite understand. I'm mostly curious because I am pretty certain that this should work on more functions than just tetration. For example, this is just a conjecture, but I think the following should hold
\( \phi^{\circ z}(\xi) = \sum_{n=0}^\infty \sum_{m=0}^n (-1)^{n-m}\binom{z}{n}_q\binom{n}{m}_qq^{\binom{n-m}{2}} \phi^{\circ m}(\xi) \)
when \( \phi(\xi_0) = \xi_0 \) and \( \phi'(\xi_0) = q \) and \( \xi \) is in a neighborhood of \( \xi_0 \). This would be a great series representation of the fractional iterate of arbitrary \( \phi \), fast converging and everything.
TY
\( \phi^{\circ z}(\xi) = \sum_{n=0}^\infty \sum_{m=0}^n (-1)^{n-m}\binom{z}{n}_q\binom{n}{m}_qq^{\binom{n-m}{2}} \phi^{\circ m}(\xi) \)
when \( \phi(\xi_0) = \xi_0 \) and \( \phi'(\xi_0) = q \) and \( \xi \) is in a neighborhood of \( \xi_0 \). This would be a great series representation of the fractional iterate of arbitrary \( \phi \), fast converging and everything.
TY