Infinite product of power series

TL;DR

This paper derives a universal formula for the coefficients of infinite products of power series with constant term 1, connecting partition and permutation structures, and unifying various binomial-type theorems.

Contribution

It introduces a new, universal coefficients formula for infinite power series products, linking partitions, permutations, and classical binomial theorems.

Findings

Provides an exact coefficients formula using partitions and permutations

Unifies multiple binomial-type theorems under a single framework

Offers new combinatorial interpretations for classical theorems

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.