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 ζ_{μ}(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 {|B_{2n}|/(2n)!}_{n≥ 1} is log-convex, where B_{n} is the n-th 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, Moll and Vignat defined the numbers a_{n}(μ)=2^{2n+1}(n+1)!(μ+1)_{n}ζ_{μ}(2n) and conjectured that the sequence {a_{n}(μ)}_{n≥ 1} is log-convex for μ=0 and μ=1. By proving that ζ_{μ}(x) is log-convex for x>1 and μ>-1, we show that the sequence {a_{n}(μ)}_{n≥1} is log-convex for any μ>-1. We introduce another function θ_{μ}(x) involving ζ_{μ}(x) and the gamma function Γ(x) and we show that logθ_{μ}(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 B_{n} 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 Download: PDF |