Closed-form expressions for the Abel functions

Determine exact formulas for the convex Abel-equation solutions A, B, I, and J defined respectively by A(x(1-x))=A(x)+1, B(x/(1+x+x^2))=B(x)+1, I(x/(1+x-x^2))=I(x)+1, and J(x(1+x)/(1+2x))=J(x)+1.

Background

The paper numerically studies four convex solutions of Abel functional equations associated with the maps x(1-x), x/(1+x+x^2), x/(1+x-x^2), and x(1+x)/(1+2x). Although asymptotic expansions and high-precision numerical values are obtained, the paper states that exact formulas are not known for any of these functions.

Closed-form Abel functions are presented as exceptional even for related elementary recurrences. Determining exact expressions would provide a symbolic description of the numerically studied functions and their associated constants.