"Natural boundary", regular tetration, and Abel matrix

[tommy1729]

Another question: is this intuitive iteration thing the same as what you get from solving the equations given by substituting a Taylor series with unknown coefficients for slog in \( \mathrm{slog}_b(b^z) = \mathrm{slog}_b(z) + 1 \) and solving the resulting system of linear equations? If so, then maybe one could also try coming at this from that direction: try plugging the coefficients for the regular slog into those equations and then seeing that they do not satisfy them (being linear, they have only one solution. If this is not \( \mathrm{rslog} \), then it should fail to satisfy them.). Just construct the Taylor series for \( \mathrm{rslog}_b(b^z) \) and see if it is the same as that for \( \mathrm{rslog}_b(z) + 1 \) for a base \( b \in [1, e^{1/e}) \).

indeed , that needs to be done.