Zeta Functions and the Log Behaviour of Combinatorial Sequences

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(is the gamma function, and we show that log θ(x) is strictly increasing for x ≥ 6. This confirms a conjecture of Sun stating that the sequence { n |B 2n |} n≥1 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 log θ µ (x) is strictly increasing for x > 8e(µ + 2) 2 . This implies that n a n (µ) < n+1 a n+1 (µ) for n > 4e(µ+2) 2 . Based on Dobinski's formula, we prove that n √ B n < n+1 √ B n+1 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 { n √ B n } n≥1 and Hölder's inequality in probability theory.

The Monotonicity and Log-Behaviour of Some Functions Related to the Euler Gamma Function

“…The derangements number d n is a classical combinatorial number. It is log-convex and ratio log-concave, see [10] and [4] respectively. Noted that {Γ(n)} n≥1 is strictly infinitely log-monotonic (see Chen et al [5]) and…”

“…Motivated by a series of conjectures of Sun [17] about the monotonicity of sequences of the forms { n √ z n }, and { n+1 √ z n+1 / n √ z n }, where {z n } n 0 is a familiar number-theoretic or combinatorial sequence, for example, the Bernoulli numbers, the Fibonacci numbers, the derangement numbers, the tangent numbers, the Euler numbers, the Schröder numbers, the Motzkin numbers, the Domb numbers, and so on. These conjectures have recently been investigated by some researchers (see [4,5,8,11,21]). The main aim of this paper is to develop some analytic techniques to deal with the monotonicity of { n √ z n } and { n+1 √ z n+1 / n √ z n } (note that the monotonicity of { n+1 √ z n+1 / n √ z n } is equivalent to the logbehaviour of { n √ z n }).…”

“…This is another motivation of this paper. In particular, the following conjecture of Chen et al [4] is still open. It is well known that B 2n+1 = 0, (−1) n−1 B 2n > 0 for n ≥ 1 and (−1) n−1 B 2n = 2(2n)!ζ(2n) (2π) 2n , see [6, (6.89)] for instance.…”

“…It is well known that B 2n+1 = 0, (−1) n−1 B 2n > 0 for n ≥ 1 and (−1) n−1 B 2n = 2(2n)!ζ(2n) (2π) 2n , see [6, (6.89)] for instance. In order to show that { n (−1) n−1 B 2n } is increasing, Chen et al [4] introduced the function θ(x) = x 2ζ(x)Γ(x + 1), where…”

Two-log-convexity of the Catalan-Larcombe-French sequence

“…With respect to the theory in this field, it should be mentioned that the log-behavior of a sequence which satisfies a three-term recurrence has been extensively studied; see Liu and Wang [9], Chen et al [2,3], Liggett [8], Došlić [5], etc.…”