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

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, $1\le m_i<n_i$ and $u_i\ne 0$ for $1\le i\le I$ 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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