The Log-Convexity of the $n$-th Root Sequence of the Distinct Partition Function

Abstract

Let $q(n)$ denote the total number of partitions of $n$ into distinct parts and $\Delta$ be the difference operator with respect to $n$. We prove that for any real number $\alpha$, there exists an integer $n(\alpha)$ such that the sequence\ $\{\sqrt[n]{q(n)/n^{\alpha } } \}_{n\ge n(\alpha) } $ is log-convex by obtaining a lower bound for $\bigtriangleup ^{2} \log\sqrt[n-1]{q(n-1)/(n-1)^\alpha }$, leading to a proof of the conjecture of Sun on the log-convexity of $\big\{\sqrt[n]{q(n)}\big\}_{n\ge46}$. Moreover, we establish an inequality on the ratio $\sqrt[n-1]{q(n-1)} / \sqrt[n]{q(n)}$ by finding an upper bound of $\Delta ^{2} \log\sqrt[n-1]{q(n-1)}$. In the end, we prove the log-balancedness of the sequences $\big\{\sqrt[n]{q(n)}\big\}_{n\ge46}$ and $\big\{\sqrt[n]{q(n)}/{n}\big\}_{n\ge46}$, which can be extended to imply that the sequence $\big\{\sqrt[n]{q(n)}/{n^\alpha}\big\}_{n\ge N(\alpha)}$ is log-balanced for any nonnegative real number $\alpha$.