tailieunhanh - Báo cáo toán học: "Unbounded regions of Infinitely Logconcave Sequences"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Unbounded regions of Infinitely Logconcave Sequences. | Unbounded regions of Infinitely Logconcave Sequences David Uminsky and Karen Yeats Department of Mathematics and Statistics Boston University Boston USA Submitted Mar 20 2007 Accepted Oct 26 2007 Published Nov 5 2007 Mathematics Subject Classifications Primary 05A10 Secondary 39B12 Abstract We study the properties of a logconcavity operator on a symmetric unimodal subset of finite sequences. In doing so we are able to prove that there is a large unbounded region in this subset that is 1-logconcave. This problem was motivated by the conjecture of Boros and Moll in 1 that the binomial coefficients are 1-logconcave. 1 Introduction In this paper we study the asymptotic behavior of the logconcavity operator on finite sequences. Before we can state the problem we will need a few definitions. We say that a sequence co Cl Cra is 1-logconcave or logconcave if ci 0 for 0 i n and c2 ci 1 Ci- 1 0 for 1 i n 1. We can extend this idea of logconcave as follows Since ci is a finite sequence of length n we define ci 0 for i 0 and i n 1 then define the operator L ci ci ci 1ci 1 1 If ci is logconcave then L ci is a new non-negative sequence. We now define a sequence ci to be if Lk ci is a non-negative sequence for all k 1. While studying a new class of integrals related to Ramanujan s Master Theorem Boros and Moll proposed that a particular family of finite sequences of coefficients di m were 1-logconcave. Boros and Moll then point out that showing that the binomial coefficients are 1-logconcave project in 1 would go a long way in showing the sequence di m is 1-logconcave. Kauers and Paule in 3 show that the di m are 1-logconcave. These conjectures motivated us to investigate the operator L on the space of finite sequences. The first author is partially supported by NSF grant DMS-0405724. Thanks to Cameron Morland for making better figures and to the referee for a very close reading. THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R72 1 Numerical experiments .