Explicit Expressions for Iterates of Power Series

TL;DR

This work develops a unified umbral-calculus framework to derive explicit formulas for both discrete and fractional iterates of invertible power series. It recovers and extends classical results (Schröder, Jabotinsky, Monkam) and introduces q-calculus to handle general derivatives at 0, yielding finite, computable expressions for iterate-coefficients and their extensions. The paper also provides robust expressions for the iterative logarithm, including Koszul-number coefficients, and demonstrates the structure and limitations of embeddability in this formal setting. Together, these results deepen the computational toolkit for iteration theory in formal power series and enhance connections between combinatorics, symmetric polynomials, and dynamical systems.

Abstract

In this paper, we present several formulas for both the discrete and fractional iterates of an invertible power series $f$, using a new unifying approach based on umbral calculus. Known formulas are extended, and their proofs simplified, while new expressions are introduced. In particular, by employing $q$-calculus identities, we eliminate the requirement for $f'(0)$ to equal $1$ and the resulting general expressions for the iterative logarithm are obtained as well.

Theorems & Definitions (23)

• theorem 1
• proof
• definition 1
• theorem 2: Monkam monkam2020
• lemma 1
• lemma 1
• proof
• proof : of \ref{['t:Monkam']}
• theorem 3: Schröder schroeder1871
• proof