Ranks of overpartitions modulo 6 and 10

Ranks of overpartitions: Asymptotics and inequalities

“…Remark 7. The identities from (1.8) and (1.9) were proved by Ji, Zhang and Zhao [16], who further proved that N (0, 6, 3n) > N (2, 6, 3n) for n ≥ 1, and N (0, 6, 3n + 1) > N (2, 6, 3n + 1) for n ≥ 0. While (1.14) follows easily now for n ≡ 0, 1 (mod 3), the inequality is not at all clear for n ≡ 2 (mod 3), as the same authors also showed that N (0, 6, 3n + 2) < N (2, 6, 3n + 2) for n ≥ 1 and N (1, 6, 3n + 2) > N (3, 6, 3n + 2) for n ≥ 0.…”

“…Remark 6. Ji, Zhang and Zhao [16] proved (1.13) for n ≡ 0 (mod 5), whereas the inequality (1.14) is new.…”

“…An outline of the proof of Theorem 1 is given in Section 2, and its proof in detail, along with that of the equidistribution of N (a, c, n), is given in Section 3. In the final section we show how to use Theorem 1 in order to prove the inequalities conjectured by Ji, Zhang and Zhao [16], and Wei and Zhang [26], which are stated in Theorems 2-4 together with some other inequalities.…”

“…Before giving the proof of Theorem 2, we must establish some identities. The following is an easy generalization of [16,Lemma 3.1]. (1 − ζ a 10 )(1 − ζ −a 10 )(−1) n q n 2 +n (1 − ζ a 10 q n )(1 − ζ −a 10 q n )…”

“…The following inequalities were conjectured by Ji, Zhang and Zhao[16, Conjecture 1.6 and Conjecture 1.7], and Wei and Zhang [26, Conjecture 5.10].…”