A Generalization of Bell Polynomials and Multinomial Expansions via Permutations on Partitions, by Perturbation expansions of Functional Determinants

Abstract

We give an exact coefficients formula of any infinite product of power series with constant term equal to $1$, by using structures from partitions of integers and permutation groups. This is an universal theorem for various of Binomial-type theorems in many sense. In particular, we give the new formulas as the double counting of Bell polynomial, Binomial Theorem and Multinomial Theorem.Comment: 17 page