📝 SummarXiv

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.