The Reality Of Zeros Of Certain Polynomials

The distribution property of combinatorial sequences is a basic problem in combinatorics. One of the most important properties is unimodality which includes unimodal, log-concave, log-convex and PF properties. The unimodal property arises often in combinatorics, algebra, geometry, probability, statistics and other branches of mathmatics. It also arises in computer science, economics and other science. Polynomials with only real zeros are also a basic problem in combinatorics. Newton inequality and Aissen-Schoenberg-Whitney theorem provide the basic link between the property of unimodality and polynomials having only real zeros. It is for a long time that people pay attention to the normal distribution of some combinatorial sequences. Normal distribution is an important distribution in probability and statistics, which is widely applied in solving practical problems. There is closely relation between polynomials with non-negative coefficients and normal distribution. Although some problems are not directly about polynomials with only real zeros, it can be inverted into polynomials with only real zeros.The organization of this paper is as follows:1. In the first chapter, we review the background and some related definitions of real zeros of certain polynomials.2. In the second chapter, we give simpler proofs for the open problems about the limiting distribution for the rows of Narayana and Fibonacci matrices proposed by Shapiro.3. In the third chapter, we give a simpler proof for a conjecture of Boros, Moll and Shallit about all zeros of certain polynomials lying on the line (?) = -1/2 in the complex plane.