Zeta Functions and the Log-behavior of Combinatorial Sequences

William Y. C. Chen, Jeremy J. F. Guo and Larry X. W. Wang

  Abstract:  In this paper, we use the Riemann zeta function ζ(x) and the Bessel zeta function ζμ(x) to study the log-behavior 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 n-th Bernoulli number. We introduce the function θ(x)=((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, Moll and Vignat defined the numbers an(μ)=22n+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(μ)}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 for n> 4e(μ+2)2. Based on Dobinski's formula, we prove that for n≥1, where Bn is the n-th Bell number. This confirms another conjecture of Sun. We also establish a connection between the increasing property of and Hölder's inequality in probability theory.

  AMS Classification:  05A20, 11B68

  Keywords:  log-convexity, Riemann zeta function, Bernoulli number, Bell number, Bessel zeta function, Narayana number, Hölder's inequality

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