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arXiv · 2501.18744

On the $q$-factorization of power series

Robert Schneider·Andrew V. Sills·Hunter Waldron
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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.

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Robert Schneider, Andrew V. Sills, Hunter Waldron. 2025-06-11. On the $q$-factorization of power series. https://doi.org/10.1007/s11139-025-01140-4

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