Turán inequalities for the broken k-diamond partition function

2022/06/20 by Dong, Janet J. W., Ji, Kathy Q., Jia, Dennis X. Q.
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2206.09512

Abstract

We obtain an asymptotic formula for Andrews and Paule's broken k-diamond partition function Δ_k(n) where k=1 or 2. Based on this asymptotic formula, we derive that Δ_k(n) satisfies the order d Turán inequalities for d≥ 1 and for sufficiently large n when k=1 and 2 by using a general result of Griffin, Ono, Rolen and Zagier. We also show that Andrews and Paule's broken k-diamond partition function Δ_k(n) is log-concave for n≥ 1 when k=1 and 2. This leads to Δ_k(a)Δ_k(b)≥Δ_k(a+b) for a,b≥ 1 when k=1 and 2.

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