Ranks of overpartitions: Asymptotics and inequalities

Abstract

In this paper we compute asymptotics for the coefficients of an infinite class of overpartition rank generating functions. Using these results, we show that $\overline{N}(a,c,n),$ the number of overpartitions of $n$ with rank congruent to $a$ modulo $c,$ is equidistributed with respect to $0\le a<c,$ for any $c\ge 2,$ as $n\to \infty$ and, in addition, we prove some inequalities between ranks of overpartitions conjectured by Ji, Zhang and Zhao (2018), and Wei and Zhang (2018) for $n=6$ and $n=10.$

23 pages; to appear in \textit{J. Math. Anal. Appl.}