On the q-factorization of power series

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发表日期：

2025年6月21日

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Robert Schneider,
Andrew V. Sills,
Hunter Waldron

摘要

Any power series with unit constant term can be factored into an infinite product of the form $\prod _{n\ge 1} (1-q^n)^{-a_n}$ $\prod_{n \geq 1} {(1 - q^{n})}^{- a_{n}}$ . We give direct formulas for the exponents $a_n$ $a_{n}$ 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.