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arXiv · 2509.10023

Some conjectures of Schlosser and Zhou on sign patterns of the coefficients of infinite products

Bing He·Linpei Li
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Abstract

Recently, Schlosser and Zhou proposed many conjectures on sign patterns of the coefficients appearing in the $q$-series expansions of the infinite Borwein product and other infinite products raised to a real power. In this paper, we will study several of these conjectures. Let \[ G(q):=\prod_{i=1}^{I}\left(\prod_{k=0}^{\infty}(1-q^{m_{i}+kn_{i}})(1-q^{-m_{i}+(k+1)n_{i}})\right)^{u_{i}} \] where $I$ is a positive integer, $1\leq m_{i}<n_{i}$ and $u_{i}\neq0$ for $1\leq i\leq I$ and $|q|<1.$ We will establish an asymptotic formula for the coefficients of $G(q)^{\delta}$ with $\delta$ being a positive real number by using the Hardy--Ramanujan--Rademacher circle method. As applications, we apply the asymptotic formula to confirm some of the conjectures of Schlosser and Zhou.

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Bing He, Linpei Li. 2025-09-12. Some conjectures of Schlosser and Zhou on sign patterns of the coefficients of infinite products. https://arxiv.org/abs/2509.10023

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