The generalization of Schr\"oder's theorem (1871): The multinomial theorem for formal power series under composition

TL;DR

This paper generalizes Schr"oder's classical theorem on formal power series composition, providing explicit formulas for coefficients in the n-fold composition when the linear term coefficient is non-zero, extending the multinomial theorem analogy.

Contribution

It introduces a new, shorter proof of Schr"oder's theorem and generalizes it for cases where the linear coefficient is non-zero, with explicit coefficient formulas.

Findings

Provides explicit formulas for coefficients in formal power series composition.

Develops a new recursive lemma sharpening Cohen's lemma.

Extends the multinomial theorem analogy to formal power series under composition.

Abstract

We consider formal power series $ f(x) = a_1 x + a_2 x^2 + \cdots $ $(a_1 \neq 0)$, with coefficients in a field. We revisit the classical subject of iteration of formal power series, the n-fold composition $f^{(n)}(x)=f(f(\cdots (f(x)\cdots))=f^{(n-1)}(f(x))=\sum\limits_{k=1}^{\infty}f_k^{(n)}x^k$ for $n=2,3,\ldots$, where $f^{(1)}(x)=f(x)$. The study of this was begun, and the coefficients $f_k^{(n)}$ where calculated assuming $a_1=1$, by Schr\"oder in 1871. The major result of this paper, Theorem 7.1.1, generalizes Schr\"oder [15]. It gives explicit formulas for the coefficients $f_k^{(n)}$ when $a_1 \neq 0$ and it is viewed as an analog to the $n^{th}$ Multinomial Theorem Under Multiplication. We prove Schr\"oder's Theorem using a new and shorter approach.The Recursion Lemma, Lemma 3.1.1,which sharpens Cohen's lemma (Lemma 2.4.2) is our key tool in the proof of Theorem 7.1.1 and it is one of the most useful recurrence relations for the composition of formal power series. Along the way we develop numerical formulas for $f_k^{(n)} {\rm where} 1\leq k\leq 5$.