The "cheta" function

[bo198214]

I recall he used something similar to my change of base formula; indeed, when I first read [1], it was a comment at the bottom of page 729 (the page headed "3. Values of Generalized Logarithms"), where it mentioned the h(x) function has a sufficient approximation after at most 5 iterations.

Walker constructs an (infinitely differentiable) auxilliary function
\( h(x)=\lim_{n\to\infty} l^{[n]}(\exp^{[n]}(x)) \), where \( l(x)=\ln(x+1) \).
That satisfies
\( h(e^x)=e^{h(x)}-1 \)

Then he composes this function with the/one regular Abel function \( g \) of \( e^x-1 \), \( g(e^x-1)=g(x)+1 \).

The resulting function \( G(x)=g(h(x)) \) satisfies
\( G(e^x)=g(h(e^x))=g(e^{h(x)}-1)=g(h(x))+1=G(x)+1 \)
i.e. is an Abel function of e^x.

So where is it similar to your change of base?