Explicit Expressions for Iterates of Power Series
Kei Beauduin

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 , 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 -calculus identities, we eliminate the requirement for to equal and the resulting general expressions for the iterative logarithm are obtained as well.
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Taxonomy
TopicsAdvanced Control Systems Optimization · Matrix Theory and Algorithms
