On a conjecture of Andrews and almost alternating sign patterns

Abstract

In this paper, we prove a sign phenomenon first observed by Andrews for certain $q$-series from Ramanujan's Lost Notebook. For three of the series considered by Andrews, namely $v_2(q)$, $v_3(q)$, and $v_4(q)$, we show that the coefficients are alternating in sign, with only a density-zero set of exceptions. Our approach yields precise asymptotic formulas for the coefficients via an adapted circle method, inspired by the work of Folsom-Males-Rolen-Storzer on the $q$-series $v_1(q)$, revealing an interplay between exponential growth and oscillatory behaviour. This interaction produces a dominant alternating sign factor, which governs the sign regularity observed numerically by Andrews. More broadly, we establish the same sign behaviour for explicit infinite families of $q$-hypergeometric series encompassing these examples, and show that it arises systematically from oscillatory asymptotics of these $q$-series near roots of unity. We introduce an additional family whose coefficients appear to exhibit similar sign regularity, suggesting that this phenomenon is widespread and may point towards a deeper underlying theory.