(01/13/2017, 08:30 PM)Vladimir Reshetnikov Wrote:Ohh! So its the convergence of the infinite sum that forces the specific value of q. I was curious as to why it wouldn't converge for other q. I was thinking there was something more complicated, but inverting the q-binomial expansion is very straightforward. Now it makes a lot of sense.(01/13/2017, 07:47 PM)JmsNxn Wrote: 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.
Because the properties of q-binomial coefficients, for natural arguments only a finite number of terms in the series are non-zero, i.e. its partial sums eventually stabilize (so the convergence is trivial). Then it is possible to prove by induction that these sums reproduce discrete sample values from which the series is built. Basically, it means that the direct q-binomial transform of a discrete sequence can be undone by the reverse q-binomial transform. The value of q does not matter here, it is only significant for convergence of the series at non-integer arguments. I can write it in more details later, if you want.
Secondly, do you have any thoughts on how to show convergence. This seems to be the only thing blocking the path. I tried using the same method I used to show convergence of the newton series, but t doesn't apply to the more general q case. No where in my attempts was I using that q is the multiplier. which causes me great discomfort. I really like this expansion, so I'm stuck figuring.