Analytic Approaches To The Log-behavior Of Sequences In Combinatorics

The main results of this thesis are some progress in the log-behavior of combinato-rial sequences, including the analytic approaches to the log-convexity of the Bernoulli numbers, the generalized Lasalle numbers, and the Bell numbers; the infinitely loga-rithmically monotonicity of the Bernoulli numbers, the Catalan numbers, and the cen-tral binomial coefficients; and the ratio log-concavity of the derangement numbers, the Motzkin numbers, the central Delannoy numbers, the Fine numbers, the number of treelike polyhexes and the Domb numbers. We also show that with a certain initial con-dition, the ratio log-concavity of the sequence{an}n≥k implies the strict log-concavity of the sequence{n√an}n≥k.This thesis consists of four chapters. The first chapter is devoted to an introduction to the background, the basic concepts and the usual notations. And we also introduce the outline of this thesis.In Chapter2, by proving the log-convexity of the Riemann zeta function ζ(x), we show that the sequence {|B2n|}n≥1is log-convex and {n√|B2n|}n≥1is strictly increas-ing, where Bn is the n-th Bernoulli number. Similarly, by proving the log-convexity of the Bessel zeta function ζu(x) for μ>-1, we show that for any μ>-1, the sequence {an(μ)}n≥1is log-convex, and n√an(μ)<n+1√an+1(μ) for n>4e(μ+2)2, where the numbers an(μ) are generalizations of Lasalle numbers. Furthermore, based on Dobin-ski’s formula, we obtain that n√Bn<n+1√Bn+1for n≥1, where Bn is the n-th Bell number. We also establish a connection between the increasing property of {n√Bn}n≥1and Holder’s inequality in probability theory.In Chapter3, based on the classical concept of continuous logarithmically com-pletely monotonic functions, we introduce the notion of infinitely logarithmically mono-tonic sequences. First, we establish the relation between the completely monotonic functions and infinitely logarithmically monotonic sequences. Then using the log-behavior of the Gamma function Γ(x) and the Riemann zeta function ζ(x), we show that the sequences of the Bernoulli numbers, the Catalan numbers and the central bino- mial coefficients are infinitely logarithmically monotonic.In Chapter4, we will point out that a logarithmically monotonic sequence{an}n≥0of order two is called ratio log-concave in the sense that the sequence{an+1/an}n≥0is log-concave. The ratio log-concavity of {an}n≥k implies that the sequence {n√an}n≥k is strictly log-concave under a certain initial condition. By finding the appropriate bound for an+1/an, we investigate the ratio log-concavity of the combinatorial se-quence{an}n≥0and show that the Motzkin numbers, the Fine numbers, the central Delannoy numbers, the number of treelike polyhexes and the Domb numbers are ratio log-concave. This implies some known results of Luca and Stanica, and Hou, Sun and Wen on the log-behavior of combinatorial sequences. Moreover, we confirm a conjec-ture of Sun on the Domb numbers, and we show this property also holds for the Catalan numbers, the central binomial coefficients, the Fine numbers and the number of treelike polyhexes. Finally, we conjecture that for any positive numbers k, the above combina-torial sequences are logarithmically monotonic of order k if we disregard several terms in front of these sequence.