Laguerre inequality and determinantal inequality for the broken k-diamond partition function

Abstract In 2007, Andrews and Paule introduced the broken k-diamond partition function ∆ k (n), which has received a lot of researches on the arithmetic propertises. In this paper, we will prove the broken k-diamond partition function satisfies the Laguerre inequalities of order 2 and the deter-minantal inequalities of order 3 for k = 1 or 2. Moreover, we conjectured the thresholds for the Laguerre inequalities of order m and the positivity of m-order determinants for 4 ≤ m ≤ 14 for the broken k-diamond partition function when k = 1 or 2. AMS Classification: 05A20, 11P82