Asymptotics and inequalities for the broken $k$-diamond partition function

Many papers have studied inequalities for Andrews and Paule's broken $k$-diamond partition function $\Delta_{k}(n)$ when $k=1$ or $2$. In this paper, we derive an exact formula for $\Delta_{k}(n)$ when $k\geq 1$. Building on this result, we also derive an asymptotic formula for $\Delta_{k}(n)$ with an explicit error bound. Using this formula, we prove that for $k\geq 1$ and sufficiently large $n$, $\Delta_{k}(n)$ satisfies the Tur\'an and Laguerre inequalities of any order and exhibits asymptotic complete monotonicity. Define $n_k:=\max\left\{\left\lceil8k^{3}+\frac{k+1}{12}\right\rceil,526\right\}$. Furthermore, we show that $\Delta_{k}(n)$ is log-concave for $k\ge3$ and $n\ge n_k$. Consequently, it follows that $\Delta_{k}(a)\Delta_{k}(b)\ge\Delta_{k}(a+b)$ for $k\ge3$ and $a,b \ge n_k$.