Abel's Functional Equation and Interrelations

TL;DR

The paper studies convex solutions of four Abel equations $A,B,I,J$, examining their interrelations via iterational asymptotics and higher-power recurrences. It presents numerical evidence for two conjectured identities $A(x/(1+x))=B(x)+1$ and $I(x/(1+x))=J(x)+1$, and compares their numerical behavior across domains; a Corrigendum later explains these identities are actually reparametrizations of the underlying recurrences. The analysis includes power–logarithmic asymptotics for iterates, detailed recurrences for ratio sequences, and reciprocal-iteration behavior that relate to hypertranscendental Abel functions, highlighting the impossibility of differential-algebraic representations. Overall, the work illuminates the structure of Abel-function dynamics, provides extensive numerical evidence, and clarifies conceptual misunderstandings via the Corrigendum.

Abstract

Convex solutions $A,B,I,J$ of four Abel equations are numerically studied. We do not know exact formulas for any of these functions, but conjecture that $A,B$ and $I,J$ are closely related. [Corrigendum at end.]

Figures (4)

• Figure 1: Blue curve is $10x(1-x)$, scaled for visibility, with maximum at $x=1/2$. Green curve is $A(x)$, with minimum at $x=1/2$. Axes are compatible.
• Figure 2: Blue curve is $10x/(1+x+x^{2})$, scaled for visibility, with maximum at $x=1$. Green curve is $B(x)$, with minimum at $x=1$. Axes are compatible.
• Figure 3: Blue curve is $10x/(1+x-x^{2})$, scaled for visibility, with inflection point at $\varphi^{1/3}-\varphi^{-1/3}=0.322$. Green curve is $I(x)$, with inflection point at $\psi=0.629$. Distance between vertical axis notches is ten times that for horizontal.
• Figure 4: Blue curve is $10x(1+x)/(1+2x)$, scaled for visibility. Green curve is $J(x)$. Distance between vertical axis notches is two times that for horizontal.