Solving tetration for base 0 < b < e^-e

[bo198214]

The problem with the real fixpoint is, that the triangular Bell-matrices have (alternating signed) units on its diagonal. This prevents the computation of a matrix-logarithm as well of the diagonalization - at least in my implementations.

That should not pose a problem. You need the first row of \( B^t \), where \( B \) is the Bell matrix. You make a Jordan decomposition \( B = S J S^{-1} \) and then \( B^t = S J^t S^{-1} \) where for each Jordanblock \( J_m \) for eigenvalue \( \lambd_m \) with multiplicity \( M_m \) one sets \( J_m^t = \sum_{n=0}^{M_m} \left(t\\n\right) (J_m-\lambda_m I)^n \). The sum is finite because \( J_m-\lambda_m I \) is nilpotent: \( J_m^{M_m}=0 \).

Unfortunately the Jordan decompostion is flawed in Sage so I could not try it myself.

But well, let's see. It's surely not the highest summit of wisdom... and we also have the Newton-binomial-formula and others...

I dont think that the Newton formula helps. It is real-valued and hence can not return a suitable solution for a decreasing base function.