关于一般素数模同余式的可解性

通过研究一般素数模整系数同余式,利用初等方法,得到同余式有解的必要条件.     

Catalan数与二项式系数和式的同余式(英文)

证明了孙智伟教授提出的猜想,它们是关于Catalan数或二阶Catalan数与二项式系数和式模奇素数p或者奇素数p平方的同余式.     

有限域上一类三项式的研究

利用有限域理论,按照扩张次数k的奇偶性,研究了p~k元域上一类三项式的可约性判定问题,并在一定的条件下给出了该类三项式的一个分解式,最后给出了两种利用此类三项式构造新的不可约多项式的方法.     

超中心扩张的根

本文引入了超中心扩张的概念，得到了类似于小心扩张的几个漂亮结果。     

第二类Stirling数的一些同余式

设k,n为非负整数,S(n,k)表示第二类Stirling数.本文研究了S(n,k)模2的方幂的同余式,首先给出了一类二项式系数模2的同余式,然后利用上述结果得到了S(n,a2~m+b)模2~m的同余式.其表达式均由简单二项式系数组成,其中m≥3,b=0,1,2.这些结果改进了Chan和Manna的结果.     

欧拉商的同余式及其应用(Ⅲ)

In the papers of 2002 and 2007, Cai et al. introduced a series of congruences involving binomial coefficients under perfect moduli. This article generalizes these congruences to cubic cases leading to many new statements. For example, the congruence Π_d|n(_└d/e┘ ^kd-1)^μ(n/d) module n³ for e=2, 3, 4 and 6, and the following congruence [Formula is presented]. © 2019, Chinese Academy of Sciences. All right reserved.     

有限域上与仿射多项式有关的一些多项式的可约性

在这篇文章中,研究了有限域上一些与仿射多项式有关的多项式的可约性.对于有限域Fp上不是xppt-x-1的仿射三项式,得到了这些三项式的一个明确的因式.完全确定了多项式g(xps-ax-b)在Fp[x]中的分解,这里g(x)是Fp[x]中一个不可约多项式.证明了Fp上次数相同的不可约多项式的全体可以构成一个正则图.同时给出了多项式g(xqs-x-b)在Fp[x]不可约因式的个数公式,这里g(x)是Fp上一个不可约多项式.     

含有中心二项式系数以及广义调和数的无穷级数恒等式

该文以两个高斯超几何求和公式为基础,建立一系列关于中心二项式系数和广义调和数的无穷级数恒等式.     

李Poisson超代数的泛中心扩张

研究了李Poisson超代数的泛中心扩张问题.通过构造其泛中心扩张,得到了其存在泛覆盖的充要条件是李Poisson超代数是完全的,并对李Poisson超代数的自同构群及导子的提升给出了结果.     

整系数多项式有理根定理的改进

文章对整系数多项式有理根定理进行了推广和改进．运用本文的定理，可使整系数多项式有理根的寻求变得简便易行．     

Congruences involving generalized central trinomial coefficients

For integers b and c the generalized central trinomial coefficient Tn(b,c)denotes the coefficient of xnin the expansion of(x2+bx+c)n.Those Tn=Tn(1,1)(n=0,1,2,...)are the usual central trinomial coefficients,and Tn(3,2)coincides with the Delannoy number Dn=n k=0n k n+k k in combinatorics.We investigate congruences involving generalized central trinomial coefficients systematically.Here are some typical results:For each n=1,2,3,...,we have n-1k=0(2k+1)Tk(b,c)2(b2-4c)n-1-k≡0(mod n2)and in particular n2|n-1k=0(2k+1)D2k;if p is an odd prime then p-1k=0T2k≡-1p(mod p)and p-1k=0D2k≡2p(mod p),where(-)denotes the Legendre symbol.We also raise several conjectures some of which involve parameters in the representations of primes by certain binary quadratic forms.     

Central polynomials for matrices over finite fields

Let c(x ₁,?…?,?x _d ) be a multihomogeneous central polynomial for the n?×?n matrix algebra M _n (K) over an infinite field K of positive characteristic p. We show that there exists a multihomogeneous polynomial c ₀(x ₁,?…?,?x _d ) of the same degree and with coefficients in the prime field 𝔽 _p which is central for the algebra M _n (F) for any (possibly finite) field F of characteristic p. The proof is elementary and uses standard combinatorial techniques only.     

两类关于二项式系数倒数的级数(英文)

By applying the theory of formal power series,the author obtains the closed forms for two kinds of infinite series involving the reciprocals of binomial coefficients,and the author gets another closed form for the infinite series Σr≥m tn＋r/（n＋rr）.     

Recurrence Relations for the Coefficients in Series Expansions with Respect to Semi-Classical Orthogonal Polynomials

Let {P _k } be a sequence of the semi-classical orthogonal polynomials. Given a function f satisfying a linear second-order differential equation with polynomial coefficients, we describe an algorithm to construct a recurrence relation satisfied by the coefficients a _k [f] in f= _k a _k [f]P _k . A systematic use of basic properties (including some nonstandard ones) of the polynomials {P _k } results in obtaining a recurrence of possibly low order. Recurrences for connection or linearization coefficients related to the first associated generalized Gegenbauer, Bessel-type and Laguerre-type polynomials are given explicitly.     

Coefficients of Wronskian Hermite polynomials

We study Wronskians of Hermite polynomials labeled by partitions and use the combinatorial concepts of cores and quotients to derive explicit expressions for their coefficients. These coefficients can be expressed in terms of the characters of irreducible representations of the symmetric group, and also in terms of hook lengths. Further, we derive the asymptotic behavior of the Wronskian Hermite polynomials when the length of the core tends to infinity, while fixing the quotient. Via this combinatorial setting, we obtain in a natural way the generalization of the correspondence between Hermite and Laguerre polynomials to Wronskian Hermite polynomials and Wronskians involving Laguerre polynomials. Lastly, we generalize most of our results to polynomials that have zeros on the p-star.     

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 .     

Laguerre-Freud equations for Generalized Hahn polynomials of type I

We derive a system of difference equations satisfied by the three-term recurrence coefficients of some families of discrete orthogonal polynomials.     

关于高阶Genocchi数和广义Lucas多项式的恒等式

利用组合数学的方法,得到了一些包含高阶Genocchi数和广义Lucas多项式的恒等式,并且由此建立了Fibonacci数与Riemann Zeta函数的关系式.