Zeta Functions and the Log-behavior of Combinatorial Sequences

Abstract

In this paper, we use the Riemann zeta function $\zeta (x)$ and the Bessel zeta function ${\zeta }_{\mu }(x)$ to study the log-behavior of combinatorial sequences. We prove that $\zeta (x)$ is log-convex for $x>1$. As a consequence, we deduce that the sequence $\{|B_{2n}|/(2n)!{\}}_{n\ge 1}$ is log-convex, where $B_n$ is the $n$-th Bernoulli number. We introduce the function $\theta (x)=(2\zeta (x)\Gamma (x+1){)}^{\frac{1}{x}}$, where $\Gamma (x)$ is the gamma function, and we show that $\log \theta (x)$ is strictly increasing for $x\ge 6$. This confirms a conjecture of Sun stating that the sequence $\{\sqrt[n]{|B_{2n}}|{\}}_{n\ge 1}$ is strictly increasing. Amdeberhan, Moll and Vignat defined the numbers $a_n(\mu )=2^{2n+1}(n+1)!(\mu +1{)}_n{\zeta }_{\mu }(2n)$ and conjectured that the sequence $\{a_n(\mu ){\}}_{n\ge 1}$ is log-convex for $\mu =0$ and $\mu =1$. By proving that ${\zeta }_{\mu }(x)$ is log-convex for $x>1$ and $\mu >-1$, we show that the sequence $\{a_n(\mu ){\}}_{n\ge 1}$ is log-convex for any $\mu >-1$. We introduce another function ${\theta }_{\mu }(x)$ involving ${\zeta }_{\mu }(x)$ and the gamma function $\Gamma (x)$ and we show that $\log {\theta }_{\mu }(x)$ is strictly increasing for $x>8e(\mu +2{)}^2$. This implies that $\sqrt[n]{a_n(\mu )}<\sqrt[n+1]{a_{n+1}(\mu )}$ for $n>4e(\mu +2{)}^2$. Based on Dobinski's formula, we prove that $\sqrt[n]{B_n}<\sqrt[n+1]{B_{n+1}}$ for $n\ge 1$, 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 $\{\sqrt[n]{B_n}{\}}_{n\ge 1}$ and Hölder's inequality in probability theory.

16 pages; to appear in Proc. Edinburgh Math. Soc. (2)

References in corpus (3)

• A probabilistic interpretation of a sequence related to Narayana polynomials
• Conjectures involving arithmetical sequences
• On some monotonic combinatorial sequences conjectures

Cited by in corpus (2)

• Higher order log-monotonicity of combinatorial sequences
• On a conjecture of Chen-Guo-Wang