Approximation of Smooth Functions and Constants by Polynomials with Integer and Natural Coefficients

Direct theorems on the approximation of smooth functions by polynomials with integer coefficients are strengthened. First theorems on approximation by polynomials with natural coefficients are obtained.     

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.     

Galois groups of polynomials arising from circulant matrices

Circulant matrices are used to construct polynomials, associated with Chebyshev polynomials of the first kind, whose roots are real and made explicit. Then the Galois groups of the polynomials are computed, giving rise to new examples of polynomials with cyclic Galois groups and Galois groups of order p(p−1) that are generated by a cycle of length p and a cycle of length p−1.     

Number of zeros of interval polynomials

In this paper, we develop a rigorous algorithm for counting the real interval zeros of polynomials with perturbed coefficients that lie within a given interval, without computing the roots of any polynomials. The result generalizes Sturm’s Theorem for counting the roots of univariate polynomials to univariate interval polynomials.     

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。     

纽结与多项式

In this paper, we deal with some corresponding relations between knots and polynomials by using the basic properties of knot polynomials （such as, some special values of knot polynomials, the Arf invariant and derivative of knot polynomials）. We give necessary and sufficient conditions that a Laurent polynomial with integer coefficients, whose breadth is less than five, is the Jones polynomial of a certain knot.     

On some generalized Sister Celine's polynomials

Certain generalizations of Sister Celine's polynomials are given which include most of the known polynomials as their special cases. Besides, generating functions and integral representations of these generalized polynomials are derived and a relation between generalized Laguerre polynomials and generalized Bateman's polynomials is established.     

The compositional inverse of a class of bilinear permutation polynomials over finite fields of characteristic 2

Laigle-Chapuy constructed a class of permutation polynomials over a finite field of characteristic 2, which includes several other known classes. In this paper, we determine the compositional inverses of all Laigle-Chapuy's permutation polynomials. Our method is based on a direct sum decomposition of the finite field.     

关于多项式多重卷积利用差分与移位算子的计值法

Here concerned and further investigated is a certain operator method for the computation of convolutions of polynomials. We provide a general formulation of the method with a refinement for certain old results, and also give some new applications to convolved sums involving several noted special polynomials. The advantage of the method using operators is illustrated with concrete examples. Finally, also presented is a brief investigation on convolution polynomials having two types of summations.     

退化高阶伯努利多项式与广义等幂和多项式的对称等式

研究了退化伯努利多项式与广义等幂和多项式的对称关系,获得了关于多个退化高阶伯努利多项式与广义等幂和多项式的若干对称关系.     

Bernoulli多项式的积分多项式

首次研究了 Bernoulli多项式的积分多项式 .首先 ,给出这类多项式的定义和基本性质 ;其次 ,建立两类幂和多项式的相互关系 ;最后 ,介绍上述结果在求解自然数幂和公式方面的应用 .     

New classes of complete permutation polynomials

In this paper, we propose several classes of complete permutation polynomials over a finite field based on certain polynomials over its subfields or subsets. In addition, a class of complete permutation trinomials with Niho exponents is studied, and the number of these complete permutation trinomials is also determined.     

A note on the coefficients of inverse polynomials

We describe some relations on the coefficients of a polynomial in terms of the map that induces and use them to characterize the coefficients of the inverse polynomials of some special classes of permutation polynomials.     

q-Laguerre polynomials and related q-partial differential equations

In this paper, we define two homogenous q-Laguerre polynomials, by introducing a modified q-differential operator, we prove that an analytic function can be expanded in terms of the q-Laguerre polynomials if and only if the function satisfies certain q-partial differential equations. Using this main result, we derive the generating functions, bilinear generating functions and mixed generating functions for the q-Laguerre polynomials and generalized q-Hahn polynomials. Cigler’s polynomials and its generating functions discussed in [J. Cao, D.-W. Niu, A note on q -difference equations for Cigler’s polynomials, J. Difference Equ. Appl. 22 (2016), 1880–1892.] are generalized. At last, we obtain an q-integral identity involving q-Laguerre polynomials. These applications indicate that the q-partial differential equation is an effective tool in studying q-Laguerre polynomials.     

Some characteristic properties of the weighted particular Schur polynomial mean

This paper provides some characteristic properties of the weighted particular Schur polynomial mean of several variables. In addition, an elementary proof of an important inequality involving the weighted particular Schur polynomial mean is given. Various related results involving a family of the Schur polynomials, symmetric polynomials, and other associated polynomials, together with the potential for their applications, are also considered.     

关于两类广义多项式的一些恒等式(英文)

By means of generating function and partial derivative methods, we investigate and establish several general summation formulas involving two classes of polynomials. The general results would apply to yield some identities for the Pell polynomials and Pell-Lucas polynomials, and other general polynomials can also be recovered in this paper.     

Several classes of complete permutation polynomials over finite fields of even characteristic

In this paper, we find three classes of complete permutation polynomials over finite fields of even characteristic. The first class of quadrinomials is complete in the sense of addition. The second and third classes of binomials and trinomials are complete in multiplication. Moreover, a result related to the complete property in multiplication of a special class of polynomials is also given.     

Multi-node higher order expansions of a function

By using the values and higher derivatives of a function at the given nodes, a kind of multi-node higher order expansion of the function is presented. The error terms of the expansions are given. Particular examples are the extensions of the Taylor polynomials, Bernstein polynomials and Lagrange interpolation polynomials. The expansions are numerical approximation polynomials and very useful particular for the functions for which the higher derivatives can be obtained easily.     

关于微分多项式零点的注记

The value distribution of differential polynomials is studied.The re- suits in this paper improve and generalize some previous theorems given by Yang Chungchun(On deficiencies of differential polynomials,Math.Z.,116(1970),197- 204),H.S.Gopalakrishna and S.S.Bhoosnurmath(On distribution of values of differential polynomials,Indian J.Pure Appl.Math.,17(1986),367-372),I.Lahiri (A note on distribution of nonhomogeneous differential polynomials,Hokkaido Math. J.,31(2002),453-458)and Yi Hongxun(On zeros of differential polynomials,Adv. in Math.,18(1989),335-351)et al.Examples show that the results in this paper are sharp.     

一元多项式学习中易犯的几个错误

结合一元多项式中的一些重要概念,如多项式的最大公因式、多项式的重根及不可约多项式等,分析一元多项式学习中易犯的错误,并强调运用定理时要注意其适用的条件和前提．