Log-convex and Stieltjes moment sequences |
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Institution: | 1. School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China;2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, PR China |
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Abstract: | 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. |
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Keywords: | Log-convex sequence Stieltjes moment sequence Totally positive matrix |
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