Two interesting sequences arose in the study of the series expansions of the complete elliptic integrals, which are called the Catalan-Larcombe-French sequence $\{P_n\}_{n\geq 0}$ and the Fennessey-Larcombe-French sequence $\{V_n\}_{n\geq 0}$ respectively. In this paper, we prove the log-convexity of $\{V_n^2-V_{n-1}V_{n+1}\}_{n\geq 2}$ and $\{n!V_n\}_{n\geq 1}$, the ratio log-concavity of $\{P_n\}_{n\geq 0}$ and the sequence $\{A_n\}_{n\geq 0}$ of Ap\'{e}ry numbers, and the ratio log-convexity of $\{V_n\}_{n\geq 1}$.

中文翻译：

---

与椭圆积分相关的两个序列的对数行为

在完全椭圆积分的级数展开的研究中出现了两个有趣的序列，它们被称为 Catalan-Larcombe-French 序列 $\{P_n\}_{n\geq 0}$ 和 Fennessey-Larcombe-French 序列 $ \{V_n\}_{n\geq 0}$ 分别。在本文中，我们证明了 $\{V_n^2-V_{n-1}V_{n+1}\}_{n\geq 2}$ 和 $\{n!V_n\}_ 的对数凸性{n\geq 1}$，$\{P_n\}_{n\geq 0}$ 的比对数凹度与Ap\'{的序列$\{A_n\}_{n\geq 0}$ e}ry 个数字，以及 $\{V_n\}_{n\geq 1}$ 的对数凸度比。