An explicit series for the tetration of a complex height

[JmsNxn]

Oh, so it's because binomials of \( q^z \) can approximate \( \phi^{\circ z}(\xi) \). That's really clever. I like that. I can't imagine how you prove this using the method I used to prove the newton series expansion. I think the best Idea possible would be to show the the n'th q-difference equation decays exponentially. I can show the n'th diference equation has exponential decay. I'm not sure how to incorporate that q is the multiplier though.