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

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 $f_k^{(n)} {\rm where} 1\leq k\leq 5$.