Explicit Expressions for Iterates of Power Series
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.
0
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.