Log-behavior of two sequences related to the elliptic integrals

Brian Y. Sun, James J. Y. Zhao

Published 2016-02-13Version 1

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 ratio log-concavity of $\{P_n\}_{n\geq 0}$, the log-convexity of $\{V_n^2-V_{n-1}V_{n+1}\}_{n\geq 2}$, the ratio log-convexity of $\{V_n\}_{n\geq 1}$, and the log-convexity of $\{n!V_n\}_{n\geq 1}$.