Now let us refocus on Walker's function, he took the Abel function \( G \) for base \( b=e \) so we generalize to arbitrary \( b \) by:
\( h_b(x)=\lim_{n\to\infty} l^{[n]} (\exp_b^{[n]}(x)) \) (though I am not sure whether it still converges for b>e. Jay, Sheldon?)
It satisfies then \( h_b(b^x)=e^{h(x)}-1 \).
Also we want to have the inverse \( G_b^{-1} = h_b^{-1} \circ g^{-1} \), where
\( h_b^{-1}(x)=\lim_{n\to\infty} \log_b^{[n]}(e^{[n]}(x)) \), \( e(x)=e^x-1 \)
and \( g^{-1} \) is a regular superfunction of \( e(x)=\exp(x)-1 \), i.e. \( g^{-1}(x+1)=e(g^{-1}(x)) \).
Together this yields
\( \operatorname{sexp}_b(x)=G_b^{-1}(x) = \lim_{n\to\infty} \log_b^{[n]}(e^{[n]}(g^{-1}(x))) = \lim_{n\to\infty} \log_b^{[n]}(g^{-1}(x+n)) \)
so we have a similar case as in the previous post but with \( T_b(x) = g^{-1}(x) \).
by modifying \( T_b \) we should get tons of variants of \( \operatorname{sexp}_b \).
\( h_b(x)=\lim_{n\to\infty} l^{[n]} (\exp_b^{[n]}(x)) \) (though I am not sure whether it still converges for b>e. Jay, Sheldon?)
It satisfies then \( h_b(b^x)=e^{h(x)}-1 \).
Also we want to have the inverse \( G_b^{-1} = h_b^{-1} \circ g^{-1} \), where
\( h_b^{-1}(x)=\lim_{n\to\infty} \log_b^{[n]}(e^{[n]}(x)) \), \( e(x)=e^x-1 \)
and \( g^{-1} \) is a regular superfunction of \( e(x)=\exp(x)-1 \), i.e. \( g^{-1}(x+1)=e(g^{-1}(x)) \).
Together this yields
\( \operatorname{sexp}_b(x)=G_b^{-1}(x) = \lim_{n\to\infty} \log_b^{[n]}(e^{[n]}(g^{-1}(x))) = \lim_{n\to\infty} \log_b^{[n]}(g^{-1}(x+n)) \)
so we have a similar case as in the previous post but with \( T_b(x) = g^{-1}(x) \).
by modifying \( T_b \) we should get tons of variants of \( \operatorname{sexp}_b \).