Sun's Log-concavity Conjecture on the Catalan–Larcombe–French Sequence

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 {P_n~(1/n)}_n≥1,which was originally conjectured by Z.W.Sun.We also obtain the strict log-concavity of the sequence {V_n~(1/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.