A positivity conjecture related first positive rank and crank moments for overpartitions

TL;DR

This paper proves a positivity conjecture related to overpartition rank and crank moments for powers of 2, and suggests it suffices to verify it for prime numbers, using a novel, simple method.

Contribution

It confirms the conjecture for all positive powers of 2 and reduces the proof to prime cases, introducing a new, simpler approach.

Findings

Proved the conjecture for all positive powers of 2.

Reduced the proof to verifying the conjecture for prime numbers.

Presented a stronger, related conjecture.

Abstract

Recently, Andrews, Chan, Kim and Osburn introduced a $q$-series $h(q)$ for the study of the first positive rank and crank moments for overpartitions. They conjectured that for all integers $m \geq 3$, \begin{equation*}\label{hqcon} \frac{1}{(q)_{\infty}} (h(q) - m h(q^{m})) \end{equation*} has positive power series coefficients for all powers of $q$. Byungchan Kim, Eunmi Kim and Jeehyeon Seo provided a combinatorial interpretation and proved it is asymptotically true by circle method. In this note, we show this conjecture is true if $m$ is any positive power of $2$, and we show that in order to prove this conjecture, it is only to prove it for all primes $m$. Moreover we give a stronger conjecture. Our method is very simple and completely different from that of Kim et al.