A Criterion for the Log-Convexity of Combinatorial Sequences
A Criterion for the Log-Convexity of Combinatorial Sequences
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DOI：
10.37236/3412
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发表时间：
2013-10
期刊：
影响因子：
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通讯作者：
Ernest X. W. Xia;Olivia X. M. Yao
中科院分区：
文献类型：
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作者：
Ernest X. W. Xia;Olivia X. M. Yao
最近，Do s Li c，Liu和Wang开发了处理序列对数凸性的技术。本文给出了某些组合序列对数凸性的一个判别准则。为了证明满足三项递归的序列的对数凸性，用我们的方法，它足以计算序列开始处的常数项。例如，为了证明Apery数$A_n$的对数凸性，通过我们的方法，我们只需要评估$A_n$对于$0\leq n \leq 6$的值。作为应用，我们证明了包括Catalan-Larcombe-French数在内的一些著名序列的对数凸性。这证实了孙的一个猜想。
Recently, Do s li c , and Liu and Wang 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 Ap e ry 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 Sun.