Log-behavior of two sequences related to the elliptic integrals

TL;DR

This paper investigates the log-behavior of sequences derived from elliptic integrals, establishing properties like log-convexity and log-concavity for several related sequences, including Catalan-Larcombe-French and Apéry numbers.

Contribution

It proves new log-behavior properties of sequences related to elliptic integrals, including log-convexity and log-concavity, extending understanding of their combinatorial and analytical structure.

Findings

Proved log-convexity of V_n^2 - V_{n-1}V_{n+1} and  n!V_n  sequences.

Established ratio log-concavity of  P_n  and  A_n  sequences.

Demonstrated ratio log-convexity of  V_n  sequence.

Abstract

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 $\{V_n\}_{n\geq 1}$.