TAILIEUCHUNG - Báo cáo toán học: "Positivity of three-term recurrence sequences"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Positivity of three-term recurrence sequences. | Positivity of three-term recurrence sequences Lily L. Liu School of Mathematical Sciences Qufu Normal University Qufu 273165 P. R. China lliulily@ Submitted Oct 10 2008 Accepted Mar 24 2010 Published Apr 5 2010 Mathematics Subject Classification 11B37 05A20 Abstract In this paper we give the sufficient conditions for the positivity of recurrence sequences defined by anun bnUn-1 CnUn-2 for n 2 where an bn cn are all nonnegative and linear in n. As applications we show the positivity of many famous combinatorial sequences. 1 Introduction The significance of the positivity to combinatorics stems from the fact that only the nonnegative integer can have a combinatorial interpretation. There has been an amount of research devoted to this topic in recent years see 1 2 5 9 10 14 15 for instance . The purpose of this paper is to present some sufficient conditions for the positivity of recurrence sequences. Let u0 u1 u2 be a sequence of integer numbers. The sequence is called a linear recurrence sequence if it satisfies a homogeneous linear recurrence relation Un aiUn-1 a2Un-2 ---- akUn-k 1 for n k where a1 a2 ak G Z. The linear recurrence relation 1 defines a unique integer sequence un n 0 after the first k initial terms u0 u1 uk-1 are given. Let p x xk a1xk-1 ak be its characteristic polynomial with discriminant D. Following 7 the positivity problem is stated as follows. Partially supported by the National Science Foundation of China under Grant . THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 R57 1 Positivity Problem. Let a linear recurrence relation 1 be given together with the initial terms ui for i 0 1 . k 1. Is the recurrence sequence un n 0 nonnegative . does it hold that un 0 for all n So far there have been some results on the positivity problem. For example Halava et al 7 presented that the positivity problem is decidable for three-term recurrence sequences defined by Un aUn-1 bUn-2 2 for a b G Z. More precisely we can conclude the .

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