Abel Functional Equation

[andydude]

Now you show that one can impose such an equation system at any development point \( x_0 \). The question is whether the resulting developments all belong to the same analytic function (which I heavily guess.)

Yes, and I can show that IF the coefficients themselves converge, THEN the infinite series (based on the exact coefficients we only have approximations of) will converge between \( x_0 \le x \le b^{x_0} \). And yes, my initial solution for the super-logarithm \( (slog_b(x)) \) was for \( x_0 = 0 \), and the combinatorial equation above is for \( x_0 = 1 \), but both should produce a single point of overlap, namely x=1.

Andrew Robbins