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November 2016 A positivity conjecture related first positive rank and crank moments for overpartitions
Xinhua Xiong
Proc. Japan Acad. Ser. A Math. Sci. 92(9): 117-120 (November 2016). DOI: 10.3792/pjaa.92.117

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*} \frac{1}{(q)_{∞}} (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. 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 completely different from that of Kim et al.

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Xinhua Xiong. "A positivity conjecture related first positive rank and crank moments for overpartitions." Proc. Japan Acad. Ser. A Math. Sci. 92 (9) 117 - 120, November 2016. https://doi.org/10.3792/pjaa.92.117

Information

Published: November 2016
First available in Project Euclid: 2 November 2016

zbMATH: 06705717
MathSciNet: MR3567597
Digital Object Identifier: 10.3792/pjaa.92.117

Subjects:
Primary: 05A17 , 11P82

Keywords: $q$-series , overpartitions , positivity

Rights: Copyright © 2016 The Japan Academy

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Vol.92 • No. 9 • November 2016
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