Two-log-convexity of the Catalan-Larcombe-French sequence

TL;DR

This paper proves that the Catalan-Larcombe-French sequence is strictly 2-log-convex by applying a criterion for 2-log-convexity to a related sequence, extending understanding of its convexity properties.

Contribution

The paper introduces a new proof that the Catalan-Larcombe-French sequence is strictly 2-log-convex using a criterion for sequences satisfying three-term recurrence relations.

Findings

The sequence is strictly 2-log-convex.

The sequence $oxed{P^2_n - P_{n-1} P_{n+1}}$ is strictly log-convex.

The sequence is log-balanced.

Abstract

The Catalan-Larcombe-French sequence $\{P_n\}_{n\geq 0}$ arises in a series expansion of the complete elliptic integral of the first kind. It has been proved that the sequence is log-balanced. In the paper, by exploring a criterion due to Chen and Xia for testing 2-log-convexity of a sequence satisfying three-term recurrence relation, we prove that the new sequence $\{P^2_n-P_{n-1}P_{n+1}\}_{n\geq 1}$ are strictly log-convex and hence the Catalan-Larcombe-French sequence is strictly 2-log-convex.