A Generalized Representation of Faá di Bruno's Formula Using Multivariate Bell Polynomials

We provide a novel representation of the total n-th derivative of the multivariate composite function $f\circ g$, i.e. a generalized Faà di Bruno’s formula. To this end, we make use of properties of the Kronecker product and the n-th derivative of the left-composite $f$, which allow the use of a multivariate form of partial Bell polynomials to represent the generalized Faà di Bruno’s formula. We further show that standard recurrence relations that hold for the univariate partial Bell polynomial also hold for the multivariate partial Bell polynomial under a simple transformation. We apply this generalization of Faà di Bruno’s formula to the computation of multivariate moments of the normal distribution.