A multivariate Faa di Bruno formula for computing arbitrary partial derivatives of a function composition is presented. It is shown, by way of a general identity, how such derivatives can also be expressed in the form of an infinite series. Applications to stochastic processes and multivariate cumulants are then delineated.

The coefficients of g(s) in expanding the rth derivative of the composite function g ◦ f by Faà di Bruno’s formula, is determined by a Diophantine linear system, which is proved to be equivalent to the problem of enumerating partitions of a finite set of integers attached to r and s canonically. © 2005 Elsevier B.V. All rights reserved. MSC: primary 11D04; secondary 05A17; 11D45; 11Y50; 15A36; ...

Using a multivariable Faa di Bruno formula we give conditions on transformations τ : [0, 1] → X where X is a closed and bounded subset of R such that f ◦ τ is of bounded variation in the sense of Hardy and Krause for all f ∈ C(X ). We give similar conditions for f◦τ to be smooth enough for scrambled net sampling to attain O(n−3/2+ ) accuracy. Some popular symmetric transformations to the simple...

A short proof of the generalized Faà di Bruno formula is given and an explicit parametrization of the set of indices involved in the coefficient of a specific term of the formula, is provided. An application is also included.

We give a new combinatorial interpretation of the noncommutative Lagrange inversion formula, more precisely, of the formula of Brouder-FrabettiKrattenthaler for the antipode of the noncommutative Faà di Bruno algebra.