Inequalities for the Broken k-Diamond Partition Function

2022/09/15 by Dennis X. Q. Jia, Jia, Dennis X. Q.
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2209.07056

openalex publication_date 2022/09/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 2007, Andrews and Paule introduced the broken k-diamond partition function Δ_k(n), which has received a lot of researches on the arithmetic propertises. In this paper, we prove that D³log Δ₁(n-1)>0 for n≥ 5 and D³ log Δ₂(n-1)>0 for n≥ 7, where D is the difference operator with respect to n. We also conjecture that for any k≥ 1 and r≥ 1, there exists a positive integer n_k(r) such that for n≥ n_k(r), (-1)^r D^r log Δ_k(n)>0. This is analogous to the positivity of finite differences of the logarithm of the partition function, which has been proved by Chen, Wang and Xie. Furthermore, we obtain that both \Δ₁(n)\_{n≥ 0} and \Δ₂(n)\_{n≥ 0} satisfy the higher order Turán inequalities for n ≥ 6.

Related

• Turán inequalities for the broken k-diamond partition function
• Laguerre inequality and determinantal inequality for the broken k-diamond partition function
• New congruences for broken k-diamond partitions
• Finite Differences of the Logarithm of the Partition Function
• Congruences for Broken k-Diamond Partitions