Multivariate bell polynomials, series, chain rules, moments and inversion

Abstract

The multivariate Bell polynomials are defined as the coefficients of a power of a multivariate series. We give recurrence relations for them for dimensions 1 to 3. We illustrate their uses in chain rules for differentiating a function of a multivariate function, relating multivariate moments and cumulants, and in multivariate iteration and inversion.