Log-convex and Stieltjes moment sequences

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• 作者：Yi Wang^a ; ^{wangyi@dlut.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Bao-Xuan Zhu^b ; ^{bxzhu@jsnu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05A20 ; 15B05 ; 15B99 ; 44A60
• 刊名：Advances in Applied Mathematics
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：81
• 期：Complete
• 页码：115-127
• 全文大小：305 K

文摘

We show that Stieltjes moment sequences are infinitely log-convex, which parallels a famous result that (finite) Pólya frequency sequences are infinitely log-concave. We introduce the concept of q-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences. We provide a criterion for linear transformations and convolutions preserving Stieltjes moment sequences. Many well-known combinatorial sequences are shown to be Stieltjes moment sequences in a unified approach and therefore infinitely log-convex, which in particular settles a conjecture of Chen and Xia about the infinite log-convexity of the Schröder numbers. We also list some interesting problems and conjectures about the log-convexity and the Stieltjes moment property of the (generalized) Apéry numbers.