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

Abstract

Many papers have studied inequalities for Andrews and Paule's broken $k$-diamond partition function $\hat{I}{”}_k(n)$ when $k=1$ or $2$. In this paper, we derive an exact formula for $\hat{I}{”}_k(n)$ when $k\ge 1$. Building on this result, we also derive an asymptotic formula for $\hat{I}{”}_k(n)$ with an explicit error bound. Using this formula, we prove that for $k\ge 1$ and sufficiently large $n$, $\hat{I}{”}_k(n)$ satisfies the Turán and Laguerre inequalities of any order and exhibits asymptotic complete monotonicity. Define $n_k:=\max \left\{\lceil 8k^3+\frac{k+1}{12}\rceil ,526\right\}$. Furthermore, we show that $\hat{I}{”}_k(n)$ is log-concave for $k\ge 3$ and $n\ge n_k$. Consequently, it follows that $\hat{I}{”}_k(a)\hat{I}{”}_k(b)\ge \hat{I}{”}_k(a+b)$ for $k\ge 3$ and $a,b\ge n_k$.

26 pages, comments are welcome