Log-behavior of two sequences related to the elliptic integrals

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≥0 and the Fennessey-Larcombe-French sequence {V n } n≥0 respectively. In this paper, we prove the log-convexity of {V 2 n − V n−1 V n+1 } n≥2 and {n!V n } n≥1 , the ratio logconcavity of {P n } n≥0 and the sequence {A n } n≥0 of Apéry numbers, and the ratio logconvexity of {V n } n≥1 .

Asymptotic r-log-convexity and P-recursive sequences

“…With the same discussion, besides the log-behavior of Catalan-Larcombe-French sequence given by Sun and Zhao [8], we find that the Catalan-Larcombe-French sequence, the Fine numbers and the Franel numbers of orders 3-6 are asymptotically r-log-convex for any integer r. This confirms the second part of the conjecture posed by Chen and Xia [3].…”