Exceptional Sign Pairs in Oscillatory Asymptotics and Conjectures of Andrews

Abstract

In this paper, we prove an \emph{exceptional sign pair phenomenon} for a general family of integer sequences with exponentially growing oscillatory asymptotics. This resolves, in particular, conjectural sign patterns proposed by Andrews in a 1986 paper for three $q$-series from Ramanujan's Lost Notebook, namely $v_2(q), v_3(q),$ and $v_4(q)$. Recent work of Kundu, Storzer, Wang, and the author established that the coefficients of these $q$-series are alternating in sign except in a density-zero set. We prove here that the exceptional indices where the alternating sign pattern fails occur infinitely often in a structured manner. More precisely, we show that there exist infinitely many pairs of consecutive coefficients having the same sign, and that the magnitude of at least one coefficient in each pair is a local minimum of the sequence of absolute values of the coefficients. Together with earlier results, this completes the resolution of Andrews' conjectures and their analogs for all the $q$-series in his original study.