Composing discrete Taylor expansions and applying Möbius inversion yields a covering-indexed Faà di Bruno formula that deforms flatly into the classical partition form for Fréchet derivatives.
Why do partitions occur in Faa di Bruno's chain rule for higher derivatives?
2 Pith papers cite this work. Polarity classification is still indexing.
2
Pith papers citing it
abstract
It is well-known that the coefficients in Faa di Bruno's chain rule for higher derivatives can be expressed via numeration of partitions. It turns out that this has a natural form as a formula for the vector case. To this formula two proofs are presented, both "explaining" its form involving partitions: one as a purely algebraic fact, and one "from first principles" for the case of Frechet derivatives of mappings between Banach spaces.
years
2026 2representative citing papers
The reduced Taylor polynomial of a composition equals the truncation of the composition of the reduced Taylor polynomials for C^k maps between Banach spaces.
citing papers explorer
-
Discrete Fa\`a di Bruno via M\"obius Inversion
Composing discrete Taylor expansions and applying Möbius inversion yields a covering-indexed Faà di Bruno formula that deforms flatly into the classical partition form for Fréchet derivatives.
-
Fa\`a di Bruno is Taylor Composition
The reduced Taylor polynomial of a composition equals the truncation of the composition of the reduced Taylor polynomials for C^k maps between Banach spaces.