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Zeta Functions and the Log Behaviour of Combinatorial Sequences

Published online by Cambridge University Press:  21 July 2015

William Y. C. Chen ,
Jeremy J. F. Guo  and
Larry X. W. Wang
William Y. C. Chen
Affiliation:
Center for Applied Mathematics, Tianjin University, Tianjin 300072, People’s Republic of China (chenyc@tju.edu.cn; guo@tju.edu.cn)
Jeremy J. F. Guo
Affiliation:
Center for Applied Mathematics, Tianjin University, Tianjin 300072, People’s Republic of China (chenyc@tju.edu.cn; guo@tju.edu.cn)
Larry X. W. Wang
Affiliation:
Center for Combinatorics, Nankai University, Tianjin 300071, People’s Republic of China (wsw82@nankai.edu.cn)
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Abstract

In this paper, we use the Riemann zeta function ζ(x) and the Bessel zeta function ζμ(x) to study the log behaviour of combinatorial sequences. We prove that ζ(x) is log-convex for x > 1. As a consequence, we deduce that the sequence {|B2n|/(2n)!}n ≥ 1 is log-convex, where Bn is the nth Bernoulli number. We introduce the function θ(x) = (2ζ(x)Γ(x + 1)) 1/x, where Γ(x) is the gamma function, and we show that logθ(x) is strictly increasing for x ≥ 6. This confirms a conjecture of Sun stating that the sequence is strictly increasing. Amdeberhan et al. defined the numbers an(μ) = 2 2n+1 (n + 1)!(μ+ 1)nζμ(2n) and conjectured that the sequence {an(μ)}n≥1 is log-convex for μ = 0 and μ = 1. By proving that ζμ(x) is log-convex for x > 1 and μ > -1, we show that the sequence {an(≥)}n>1 is log-convex for any μ > - 1. We introduce another function θμ,(x) involving ζμ(x) and the gamma function Γ(x) and we show that logθμ(x) is strictly increasing for x > 8e(μ + 2)2. This implies that

Based on Dobinski’s formula, we prove that

where Bn is the nth Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property of and Holder’s inequality in probability theory.

Keywords

log-convexityRiemann zeta functionBernoulli numberBell numberBessel zeta functionNarayana numberPrimary 05A2011B68
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 3 , October 2015 , pp. 637 - 651
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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