On the $q$-factorization of power series

TL;DR

This paper explores the $q$-factorization of power series with unit constant term, providing explicit formulas for the exponents in the infinite product representation and connecting them to partition enumeration functions.

Contribution

It introduces direct formulas linking power series coefficients to $q$-factorization exponents and establishes identities for partition enumeration functions.

Findings

Formulas for exponents $a_n$ in terms of power series coefficients

Identities for partition enumeration functions

$q$-analogues of enumeration formulas

Abstract

Any power series with unit constant term can be factored into an infinite product of the form $\prod_{n\geq 1} (1-q^n)^{-a_n}$. We give direct formulas for the exponents $a_n$ in terms of the coefficients of the power series, and vice versa, as sums over partitions. As examples, we prove identities for certain partition enumeration functions. Finally, we note $q$-analogues of our enumeration formulas.