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 e-prints
- Pub Date:
- May 2016
- DOI:
- 10.48550/arXiv.1605.09135
- arXiv:
- arXiv:1605.09135
- Bibcode:
- 2016arXiv160509135X
- Keywords:
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- Mathematics - Number Theory;
- 11P82;
- 05A30