Two interesting sequences arose in the study of the series expansions of the complete elliptic integrals,which are called the Catalan-Larcombe-French sequence{Pn}n≥0 and the Fennessey-Larcombe-French sequence{Vn}n≥0...Two interesting sequences arose in the study of the series expansions of the complete elliptic integrals,which are called the Catalan-Larcombe-French sequence{Pn}n≥0 and the Fennessey-Larcombe-French sequence{Vn}n≥0 respectively.In this paper,we first establish some criteria for determining log-behavior of a sequence based on its three-term recurrence.Then we prove the log-convexity of{Vn^2-V(n-1)V(n+1)}n≥2 and{n!Vn}n≥1,the ratio log-concavity of{Pn}n≥0 and the sequence{An}n≥0 of Apéry numbers,and the ratio log-convexity of{Vn}n≥1.展开更多
基金partially supported by the National Science Foundation of Xinjiang Uygur Autonomous Region(No. 2017D01C084)the National Science Foundation of China (Nos. 11771330 and 11701491)
文摘Two interesting sequences arose in the study of the series expansions of the complete elliptic integrals,which are called the Catalan-Larcombe-French sequence{Pn}n≥0 and the Fennessey-Larcombe-French sequence{Vn}n≥0 respectively.In this paper,we first establish some criteria for determining log-behavior of a sequence based on its three-term recurrence.Then we prove the log-convexity of{Vn^2-V(n-1)V(n+1)}n≥2 and{n!Vn}n≥1,the ratio log-concavity of{Pn}n≥0 and the sequence{An}n≥0 of Apéry numbers,and the ratio log-convexity of{Vn}n≥1.
