The Combinatorial Properties Of The Boros-Moll Sequences

In this thesis, we aim to show that the Boros-Moll sequences and some other combinatorial sequences which have three-term recurrence relations possess some important combinatorial properties. We show that the Boros-Moll sequences satisfy the ratio monotone property and the interlacing log-concavity, which are first defined by us. We also confirm a conjecture made by Moll and solve an open problem given by Kauers and Paule. Moreover, we present a uniform approach to show that some combinatorial sequences having three-term recurrence relations are 2-log-convex. Some conjectures are also presented in this thesis.The first chapter is devoted to the background knowledge on the Boros-Moll sequences. In their study of a quartic integral, Boros and Moll discovered a sequence{di}i=0m defined by which is called the Boros-Moll sequence. We also present the background knowl-edge of the Aprey numbers and some basic definitions that are used throughout the thesis.In Chapter 2, we give the definition of the ratio monotone property and show that the Boros-Moll sequences possess the ratio monotone property which implies the log-concavity and the spiral property. Thus, our result is stronger than the log-concavity of the Boros-Moll sequences which was conjectured by Moll and confirmed by Kauers and Paule.In Chapter 3, we present the definition of the interlacing log-concavity and prove that the Boros-Moll sequences satisfy the interlacing log-concavity which implies the log-concavity. We also show that the sequence{di(m)}m=i∞is log-concave for i≥1 and the sequence{d0(m)}m=∞is log-convex. In Chapter 4, we give a proof of Moll's minimum conjecture which states that the sequence{i(i+1)(di2(m)-di-1(m)di+1(m))}i=1m attains its minimum at i=m with 2-2mm(m+1)(m2m)2. This conjecture is stronger than the log-concave conjecture. The proof mainly relies on the spiral property of the Boros-Moll sequences and the log-concavity of the sequence{i!di(m)}i=0m.In Chapter 5, the 2-log-concavity of the Boros-Moll sequences is established, that is for 1≤i≤m-1, (di-12(m)-di-2(m)di(m))(di+12(m)-di(m)di+2(m))<(di2(m)-di-1(m)di+1(m))2, which is an open problem given by Kauers and Paule. Our main idea is to find a function f(m, k) such that As an application of the 2-log-concavity, we shall give an alternative proof of Moll's minimum conjecture.In Chapter 6, we present a uniform approach to proving the 2-log-convexity of sequences satisfying three-term recurrence relations. By this method, we show that the Apery numbers, the Cohen-Rhin numbers, the Motzkin numbers, the Fine numbers, the Franel numbers of order 3 and 4 and the large Schroder num-bers are all 2-log-convex.