Is this entire expression for tetration legal

[andydude]

\( (^\omega b)^s = \sum_{N=0}^{\infty} \frac{\sum_{k=0}^{N} \frac{(-1)^{N-k}}{(N-k)!}(^k b)^s}{\Gamma(\omega - N +1)} \)

Mindblowing.

I'll have to look at it a bit more, but this method reminds me a lot of Woon's approach, which many on this forum have lumped in with Newton.

Code:

JmsTetrate[b_, w_, s_, m_] := Sum[Sum[(-1)^(n-k)*Tetrate[b, k]^s/(n-k)!, {k, 0, n}]/Gamma[w - n + 1], {n, 0, m}];

I tested your method with the above snippet, and it works well for small m, but it doesn't take long to blow up and overflow/underflow. I don't remember if I read it on this forum or not, but I believe someone has researched which bases the Newton methods work well for, and perhaps I was using the wrong bases. Anyway, this is certainly an interesting approach, even if it is a spin on something known.

Andrew Robbins

Refs:
S.C.Woon "Analytic Continuation of Operators" http://arxiv.org/abs/hep-th/9707206