Z／mZ上的多元置换多项式

本文研究了一类典型的模ｐ的多元奇异多项式，得到了它们是模Ｐｌ（ｌ＞１）的置换多项式的充要条件并给出了一个是模ｐ２的置换多项式但不是模ｐ３的置换多项式的多元多项式例子，从而说明模ｐｌ（ｌ＞１）的多元置换多项式不能（象一元那样）简化到模ｐ上．     

有限域上一类新的置换多项式

由于有限域上的置换多项式在密码、编码和组合设计有着重要的应用,置换多项式是人们比较感兴趣的一个研究课题.利用线性化多项式,得到了一类新的形如(x~(p~k)-x+δ)~s+L(x)的置换多项式.     

有限域上一类复合对合多项式的存在性

本文作者在2021年给出了有限域上两个对合多项式的复合多项式也是对合多项式的充要条件.由于对合多项式是一类特殊的置换多项式,本文基于多项式的不动点集合与非不动点集合的关系,得到F_q上对合多项式与置换多项式复合后也是对合的充要条件.     

对称多项式概念的推广

定义一类G-对称多项式。它是对于Sn的子群G中置换不变的多项式,当G为Sn或一个n轮换生成的循环子群时,相应的G-对称多项式就是对称多项式或轮换对称多项式。     

Z/plZ上一类典型的多元奇异置换多项式

本文得到了一类典型模p奇异的n元多项式是模pl置换多项式的一个充分必要条件.特别地,对任意正整数u＞1,本文得到了一个模pu的但不是pu+1的置换多项式.这些结果是对张起帆、胡永忠等人的若干已知结果的推广和改进.     

Z/plZ上一类典型的多元奇异置换多项式

本文得到了一类典型模p奇异的n元多项式是模pl置换多项式的一个充分必要条件.特别地,对任意正整数u＞1,本文得到了一个模pu的但不是pu+1的置换多项式.这些结果是对张起帆、胡永忠等人的若干已知结果的推广和改进.     

Z/pkZ上的多元置换多项式

设Z表整数集,p为一给定奇素数,k为一正整数.张起帆(1995)得到了一类模p奇异的二元多项式成为剩余类环Z/pkZ上的置换多项式的一个充要条件,胡永忠(2001)将张起帆的这一结果的充分条件推广到了一般n元的情形.本文得到了一类模p奇异的n元多项式成为剩余类环Z/pkZ上的置换多项式的一个充要条件,所得结论是对张文(n=2)和胡文(n＞2)的自然推广和改进.     

置换多项式$x^{1+\frac{q-1}{m}}+ax$

设$m$为正整数, $F_{q^r}$是特征为$p$的有限域. 本文证明了如果$p>m^2-m$且$q\equiv 1\pmod{m}$, 则多项式$x^{1+\frac{q-1}{m}}+ax~(a\neq0)$不是$F_{q^r}~(r\geq2)$上的置换多项式. 本文还证明了$q\equiv 1\pmod{7}$且$p\neq 2, 3$时, $x^{1+\frac{q-1}{7}}+ax~(a\neq0)$不是$F_{q^r}~(r\geq2)$上的置换多项式     

F_2~n上4次正形置换多项式的形式与计数

分组密码是现代密码学中一个重要的研究分支,而置换理论在分组密码中有重要的地位.1995年,美国Teledyne电子技术公司的Lothrop Mittenthal博士提出了一种置换,即正形置换.正形置换是一类完全映射,完全映射是由Mann在1942年研究正交拉丁方的构造时引入的,其具有良好的密码学性质(良好的扩散性和完全平衡性),因此,正形置换常用来构造密码系统的算法,研究正形置换也就非常有必要.本文根据文章[1]的方法讨论了F2n(n=4,5)上的4次正形置换多项式的形式与计数,至于n5的情形我们将在以后的篇章中继续讨论.     

F2^n上4次正形置换多项式的形式与计数

分组峦码是现代密码学中一个重要的研究分支，而置换理论在分组密码中有重要的地位．199j年，美国Tcledyne电子技术公司的Lothrop Mittenthal博士提出了一种置换，即正形置换．止形置换是一类完全映射，完全映射是由Mann在1942年研究正交拉丁方的构造时引入的，其具有良好的密码学性质（良好的扩散性和完令平衡性），因此，正形置换常用来构造密码系统的算法，研究正形置换也就非常订必要．本文根据文章[1]的方法讨论了F2^n（n=4，5）上的4次正形置换多项式的形式与计数，至于n〉5的情形我们将在以后的篇章中继续讨论．     

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.     

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.     

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.     

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.     

Some generalized hypergeometric d-orthogonal polynomial sets

In this paper, we characterize the d-orthogonal polynomial sets given by their explicit expressions in a specific basis. As application, we consider the generalized hypergeometric case to characterize d-orthogonal polynomial sets of Laguerre type, Meixner type, Meixner-Pollaczek type, Krawtchouk type, continuous dual Hahn type, and dual Hahn type. For d=1, we obtain a unification of some characterization theorems in the orthogonal polynomials theory.     

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.