关于Lucas多项式平方和的恒等式

利用广义Lucas多项式L n(x,y)的性质,通过构造组合和式T n(x,y;tx2),结合Bernoulli多项式的生成函数和Euler多项式的生成函数,采用分析学中的方法,得到两个有关L2n(x,y)的恒等式.并从这一结果出发,得到了两个推论,推广了相关文献的一些结果.     

确定有限域上给定周期的不可约多项式的个数以及利用低次不可约多项式构造高次不可约多项式

主要利用较献[4]更为简明的方法证明了有关有限域Fq(q为一个素数幂）上的以l为周期的n次不可约多项式的个数的结论。另外，本结合结合初等数论知识得到了前面这个结论的几个推论，并对利用低次不可约多项式构造高次不可约多项式进行了研究。     

关于包含Hermite-Laguerre多项式的恒等式

通过讨论一类函数的高阶导数 ,建立了一些包含 Hermite-Laguerre多项式的恒等式 ,推广了著名的 Cauchy-Sheehan组合恒等式 .     

矩阵多项式的逆矩阵的求法

给出了矩阵多项式的逆矩阵的一般求法.     

多项式代数的一类新的试验多项式 *

域上多项式代数K[ X ]中的一个多项式p称为试验多项式 ,如果代数K[ X]的每个固定p的自同态必为自同构 .给出了一类新的试验多项式 ,可识别多项式代数的非线性自同构 .对于域K的特征为奇素数的情形 ,构作了可识别非半单自同构的试验多项式 .     

多元整系数多项式因式分解的一种理论和算法

本文将ｔ（ｔ是大于２的整数）元整系数多项式看成为系数为ｔ－２元整系数多项式的二元多项式，建立了多元整系数多项式因式分解的一种新理论，进而得到了分解多元整系数多项式的一个有力的算法。     

二元整系数多项式因式分解的一种理论和算法

我们发现可以把二元多项式盾成系数为一元多项式的一元多项式来进行分解，据此，本文建立了二元整系数多项式因式分解的一种理论，提出了一个完整的分解二元整系数多项式的算法。这个算法还能很自然地推广成分解多元整系数多项式的算法。     

最简型的Hermite插指

本文提出了Hermite插值问题的一种新形式，幂指数形式，简称Hermite插指。     

推广的Kantorovich多项式的一些基本性质

本文对推广的Bernstein-Kantorovich多项式进行了深入地研究和讨论,给出并证明了一些重要的基本性质.     

整系数多项式在整数环上不可约性的进一步探讨

本文首先给出了整系数多项式有二次整系数多项式因式的一个必要条件，进而通过对整系数多项式ｆ（ｘ）＝ＡｎＸ２十αｎ－１Ｘｎ－１＋…＋αｏ中ｘｎ－２的系数αｎ－２的讨论，得到一类整系数多项式在整数环上是否可约的一个判别法。     

Matrix exceptional Laguerre polynomials

We give an analog of exceptional polynomials in the matrix-valued setting by considering suitable factorizations of a given second-order differential operator and performing Darboux transformations. Orthogonality and density of the exceptional sequence are discussed in detail. We give an example of matrix-valued exceptional Laguerre polynomials of arbitrary size.     

On modified Chebyshev polynomials

In this paper some new properties and applications of modified Chebyshev polynomials and Morgan-Voyce polynomials will be presented. The aim of the paper is to complete the knowledge about all of these types of polynomials.     

On zeros of discrete orthogonal polynomials

We exploit difference equations to establish sharp inequalities on the extreme zeros of the classical discrete orthogonal polynomials, Charlier, Krawtchouk, Meixner and Hahn. We also provide lower bounds on the minimal distance between their consecutive zeros.     

Bivariate fibonacci like p-polynomials

In this article, we study the bivariate Fibonacci and Lucas p-polynomials (p ? 0 is integer) from which, specifying x, y and p, bivariate Fibonacci and Lucas polynomials, bivariate Pell and Pell-Lucas polynomials, Jacobsthal and Jacobsthal-Lucas polynomials, Fibonacci and Lucas p-polynomials, Fibonacci and Lucas p-numbers, Pell and Pell-Lucas p-numbers and Chebyshev polynomials of the first and second kind, are obtained. Afterwards, we obtain some properties of the bivariate Fibonacci and Lucas p-polynomials.     

Simultaneous Preservation of Orthogonality of Polynomials by Linear Operators Arising from Dilation of Orthogonal Polynomial Systems

For an orthogonal polynomial system and a sequence of nonzero numbers,let be the linear operator defined on the linear spaceof all polynomials via for all .We investigate conditions on and under which can simultaneously preserve the orthogonality ofdifferent polynomial systems. As an application, we get that for , a generalized Laguerre polynomial system, no can simultaneously preserve the orthogonality of twoadditional Laguerre systems, and , where and . On the other hand, for ,the Chebyshev polynomial system and , simultaneously preserves the orthogonality of uncountablymany kernel polynomial systems associated with p. We study manyother examples of this type.     

On some special polynomials

New special functions called -functions are introduced. Connections of -functions with the known Legendre, Chebyshev and Gegenbauer polynomials are given. For -functions the Rodrigues formula is obtained.     

Some Classical Polynomials Seen from Another Side

The sequence of orthogonal polynomials is said to be classical if is also orthogonal. The aim of this paper is to find the sequences which have the property that is also orthogonal. We prove that sequences, with this property have to be, classical and belong either to the set of Laguerre or Jacobi polynomials, where in the Laguerre case c has to be zero and in the Jacobi case c = ±1.     

Permutation polynomials of the form

Recently, several classes of permutation polynomials of the form (x²+x+δ)^s+x over have been discovered. They are related to Kloosterman sums. In this paper, the permutation behavior of polynomials of the form (x^p−x+δ)^s+L(x) over is investigated, where L(x) is a linearized polynomial with coefficients in . Six classes of permutation polynomials on are derived. Three classes of permutation polynomials over are also presented.