A generalized representation of Faà Di Bruno'S formula using multivariate and matrix‐valued Bell polynomials

Abstract

We provide a generalization of Faà di Bruno’s formula to represent the 𝑛-th total derivative of the multivariate and vector-valued composite 𝑓 ∘𝑔. 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 and matrix-valued 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.