Sun’s log-concavity conjecture on the Catalan–Larcombe–French sequence

Abstract

Let {P n } n≥0 denote the Catalan-Larcombe-French sequence, which naturally came from the series expansion of the complete elliptic integral of the first kind. In this paper, we prove the strict log-concavity of the sequence { n √ P n } n≥1 , which was originally conjectured by Z. W. Sun. We also obtain the strict log-concavity of the sequence { n √ V n } n≥1 , where {V n } n≥0 is the Fennessey-LarcombeFrench sequence arising from the series expansion of the complete elliptic integral of the second kind.

Log-behavior of two sequences related to the elliptic integrals

“…strict log-convexity) of the sequence { n √ S n } n≥N under an initial condition [4, Theorems 3.1 & 3.6]. Although the strictly log-concavity of { n √ P n } n≥1 and { n √ V n } n≥1 had been proved by Zhao [20] in a direct way, the ratio log-behaviors of {P n } n≥0 and {V n } n≥1 still deserve attention, and are precisely described as follows.…”

On a ratio monotonicity conjecture of a new kind of numbers

“…In recent years, Sun [ 2 , 3 ] posed a series of conjectures on monotonicity of sequences of the forms , , and . It is worth mentioning that many scholars have made valuable progress on this topic, such as Chen et al [ 4 ], Hou et al [ 5 ], Luca and Stănică [ 6 ], Wang an Zhu [ 1 ], Sun et al [ 7 ], and Zhao [ 8 ].…”

On a conjecture of Chen-Guo-Wang

“…strictly log-convex ) if the corresponding inequality is strict. Sun's conjectures stimulated a large number of further works on this topic, see [3,4,5,6,9,10,11,12].…”