A Generalized Representation of Fa\'a di Bruno's Formula Using Multivariate Bell Polynomials
Classical Analysis and ODEs
2023-12-19 v1
Abstract
We provide a novel representation of the total n-th derivative of the multivariate composite function , i.e. a generalized Fa\`a 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 , which allow the use of a multivariate form of partial Bell polynomials to represent the generalized Fa\`a 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 Fa\`a di Bruno's formula to the computation of multivariate moments of the normal distribution.
Cite
@article{arxiv.2312.10491,
title = {A Generalized Representation of Fa\'a di Bruno's Formula Using Multivariate Bell Polynomials},
author = {Michael P. Evers and Markus Kontny},
journal= {arXiv preprint arXiv:2312.10491},
year = {2023}
}