The 2-log-convexity of the Apéry Numbers
William Y.C. Chen and Ernest X. W. Xia
Abstract: We present an approach to proving the 2-log-convexity of sequences satisfying three-term recurrence relations. We show that the Apéry numbers, the Cohen-Rhin numbers, the Motzkin numbers, the Fine numbers, the Franel numbers of order 3 and 4 and the large Schröder numbers are all 2-log-convex. Numerical evidence suggests that all these sequences are k-logconvex for any k ≥ 1 possibly except for a constant number of terms at the beginning.
AMS Classification: 05A20; 11B37, 11B83
Keywords: Apéry numbers, log-convexity, 2-log-convexity, infinite log-convexity.