Fa\`a di Bruno is Taylor Composition

Abstract

We approach Fa\`a di Bruno as a composition theorem for Taylor polynomials. For $C^k$ maps $\phi: E \to F$ and $\psi: F \to G$ between Banach spaces, let $T^k_\ast(\phi; x)$ denote the reduced Taylor polynomial of $\phi$ at $x$, obtained by removing the constant term. We show that $$T^k_\ast(\psi \circ \phi; x) = \pi_{\le k}\bigl(T^k_\ast(\psi; \phi(x)) \circ T^k_\ast(\phi; x)\bigr).$$ The proof is an elementary estimate of the Peano remainder and does not use partitions or combinatorial enumeration. Expanding this composition identity recovers the classical Fa\`a di Bruno formulas. Polarization gives the multivariate partition formula (L\'evy 2006), while coefficient extraction gives the multi-index formula (Constantine and Savits 1996). Our approach separates the functorial nature of Taylor approximation from the combinatorial bookkeeping of polarization and coefficient extraction. As an application, we give a general higher-order product rule.