Is this entire expression for tetration legal

[tommy1729]

Being quick I'll write the glorious formula; \( \forall b \in \mathbb{R}\,\,;\,\, b > e^{\frac{1}{e}} \):

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

considering the equation above ; if we do \( exp(-1/x) \) on both sides that should be equivalent to adding 1 to \( \omega \).

If that is not the case then the fundamental functional equation of tetration is not satisfied.

So can you confirm that doing \( exp(-1/x) \) on both sides does what it is suppose to do ?

For those confused notice \( exp(-1/x) = 1/exp(1/x) \) and \( 1/exp^{[n]}(1/x) = (1/exp(1/x))^{[n]}. \)

regards

tommy1729