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

arXiv eprints
 Pub Date:
 May 2016
 arXiv:
 arXiv:1605.09135
 Bibcode:
 2016arXiv160509135X
 Keywords:

 Mathematics  Number Theory;
 11P82;
 05A30