Multivariate Bell polynomials and their applications to powers and fractionary iterates of vector power series and to partial derivatives of composite vector functions

If a power series is raised to a power, the coefficients of the new power series are called the partial Bell polynomials. They are useful for obtaining the inverse power series, iterations (possibly fractional) of the original power series, logs and exponents of power series, the derivatives of a composite function, and for solving various problems in partition theory. Multivariate versions of these diverse and useful results are given in terms of multivariate Bell polynomials, introduced here for the first time.