A criterion for the log-convexity of combinatorial sequences
Summary: Recently, \textit{T. Došlić} [Int. J. Math. Math. Sci. 2005, No. 4, 507--522 (2005; Zbl 1073.05005); J. Math. Inequal. 3, No. 3, 437--442 (2009; Zbl 1230.05046)], and \textit{L. L. Liu} and \textit{Y. Wang} [Adv. Appl. Math. 39, No. 4, 453--476 (2007; Zbl 1131.05010)] developed techniques for dealing with the log-convexity of sequences. In this paper, we present a criterion for the log-convexity of some combinatorial sequences. In order to prove the log-convexity of a sequence satisfying a three-term recurrence, by our method, it suffices to compute a constant number of terms at the beginning of the sequence. For example, in order to prove the log-convexity of the Apéry numbers \(A_n\), by our method, we just need to evaluate the values of \(A_n\) for \(0\leq n \leq 6\). As applications, we prove the log-convexity of some famous sequences including the Catalan-Larcombe-French numbers. This confirms a conjecture given by \textit{Z. W. Sun} [Ser. Number Theory Appl. 8, 244--258 (2013; Zbl 1308.11031)].
scientific article; zbMATH DE number 6330334
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