## A positivity conjecture related first positive rank and crank moments for overpartitions.(English)Zbl 1415.11136

Summary: Recently, G. Andrews et al. [Ann. Comb. 20, No. 2, 193–207 (2016; Zbl 1405.11136)] 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 $$m \geq 3$$, $\frac{1}{(q)_{\infty}} (h(q) - m h(q^{m}))$ has positive power series coefficients for all powers of $$q$$. B. Kim et al. [Arch. Math. 102, No. 4, 357–368 (2014; Zbl 1305.11086)] 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. [loc. cit.].

### MSC:

 11P82 Analytic theory of partitions 05A17 Combinatorial aspects of partitions of integers

### Keywords:

overpartitions; $$q$$-series; positivity

### Citations:

Zbl 1405.11136; Zbl 1305.11086
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