Combinatorial analysis of integer power product expansions

Abstract

Let $f(x)=1+\sum _{n=1}^{\mathrm{\infty }}{a}_n{x}^n$ be a formal power series with complex coefficients. Let $\{r_n{\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of nonzero integers. The Integer Power Product Expansion of $f(x)$, denoted ZPPE, is $\prod _{k=1}^{\mathrm{\infty }}(1+w_k{x}^k{)}^{r_k}$. Integer Power Product Expansions enumerate partitions of multi-sets. The coefficients $\{w_k{\}}_{k=1}^{\mathrm{\infty }}$ themselves possess interesting algebraic structure. This algebraic structure provides a lower bound for the radius of convergence of the ZPPE and provides an asymptotic bound for the weights associated with the multi-sets.