Explicit Expressions for Iterates of Power Series

TL;DR

This paper introduces new formulas for iterates of invertible power series, unifying and extending existing results through umbral calculus and q-calculus, including fractional and iterative logarithms.

Contribution

It provides a unified approach to derive explicit formulas for both discrete and fractional iterates of power series, extending known results and simplifying proofs.

Findings

Extended formulas for power series iterates

Eliminated restrictions on derivative at zero using q-calculus

Derived explicit expressions for iterative logarithms

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.