E0026 corollary notation E0020 → E0001

Classical Faà di Bruno formulas

The Fréchet iterated Faà di Bruno E0022 specializes to the classical multi-index formula of Constantine–Savits and the univariate Bell polynomial form.

(1) Corollary (Constantine–Savits). For two \(C^n\) maps \(g: X \to Y\), \(f: Y \to Z\) between Banach spaces, \(x \in X\), \(y = g(x)\), and \(\gamma \in \IN_0^k\) with \(1 \leq |\gamma| \leq n\):

\[ D(f \circ g;\, x;\, v_\bullet^{\times \gamma}) = \sum_{\kappa \vdash \gamma} \frac{\gamma!}{\kappa!\, \prod_\alpha (\alpha!)^{\kappa(\alpha)}}\, D(f;\, y;\, (D(g;\, x;\, v_\bullet^{\times \alpha})) _\alpha^{\times \kappa(\alpha)}). \]

The sum runs over multisets \(\kappa\) of nonzero multi-indices with \(\sum_\beta \kappa(\beta)\, \beta = \gamma\). The coefficient is the partition grouping coefficient \(\Part_2(\gamma, \kappa) = \gamma! / (\kappa! \prod_\alpha (\alpha!)^{\kappa(\alpha)})\).

Proof. Set \(m = 2\) in the Fréchet iterated Faà di Bruno E0022. The iterated multi-index \(\kappa \in \KM_+^2(k)\) is a multiset of nonzero multi-indices with \(\lf(\kappa) = \gamma\), i.e. \(\sum_\alpha \kappa(\alpha) \alpha = \gamma\). The iterated differential \(D^\kappa(g, f; x; v_\bullet) = D^{|\kappa|}(f; y; (D^\alpha(g; x; v_\bullet)) _\alpha^{\times \kappa(\alpha)})\). The partition grouping coefficient is \(\Part_2(\gamma, \kappa) = \gamma! / (\kappa! \prod_\alpha (\alpha!)^{\kappa(\alpha)})\), which counts the number of partitions of \(S(\gamma)\) with block-type profile \(\kappa\).

(2) Corollary (Univariate Faà di Bruno). For \(C^n\) functions \(f, g: \IR \to \IR\), the Constantine–Savits formula with \(k = 1\) and \(\gamma = n\) gives:

\[ (f \circ g)^{(n)}(x) = \sum_{\substack{k_1 + 2k_2 + \cdots + nk_n = n \\ k_j \geq 0}} \frac{n!}{k_1!\, k_2! \cdots k_n!}\, f^{(k_1 + \cdots + k_n)}(g(x)) \prod_{j=1}^n \left(\frac{g^{(j)}(x)}{j!}\right)^{k_j}. \]

This is the classical Faà di Bruno formula in Bell polynomial form.

Proof. Set \(k = 1\) in the Constantine–Savits formula. A multiset \(\kappa\) of positive integers with \(\sum_j j \kappa(j) = n\) corresponds to a tuple \((k_1, \dots, k_n)\) with \(k_j = \kappa(j)\) and \(\sum j k_j = n\). The \(\alpha\)-th derivative of a univariate map is \(D^\alpha(g; x; v) = g^{(\alpha)}(x) v^\alpha\), so \(D(g; x; v^{\times \alpha}) = g^{(\alpha)}(x) v^\alpha\). Setting \(v = 1\) and substituting gives the stated formula; the coefficient \(\gamma! / (\kappa! \prod (\alpha!)^{\kappa(\alpha)})\) becomes \(n! / \prod_j k_j!\) after absorbing the \((j!)^{k_j}\) into the derivative terms.

(3) Validation (AI review, 2026-07-19, claude-fable-5, pass). Both specializations checked against E0022 (\(m = 2\) instantiation; univariate coefficient identity \(n!/(\prod_j k_j!\, (j!)^{k_j})\) matches the Bell form with \((j!)^{k_j}\) absorbed). Added the missing hypothesis \(1 \leq |\gamma|\) (at \(\gamma = 0\) the sum is empty but the left side is \((f \circ g)(x)\)); the univariate form is unaffected. No other issues found.