浅谈二项式定理的基本应用

二项式定理实际上是初中学习的多项式乘法的继续,是解决某些整除性、近似计算问题的一种方法,与概率理论中的二项分布有其内在联系,是学习概率知识以及进一步学习概率统计的准备知识,二项式系数都是一些特殊的组合数,利用二项式定理可得到关于组合数的一些恒等式,从而深化对组合数的认识．下面我就二项式定理的应用谈几个问题：     

组合数公式的变形与组合数数列求和

组合数公式的变形与组合数数列求和胡岩火（浙江省永康二中３２１３００）高中代数课本第三册介绍了组合数公式：以及直接利用它求值、证明简单恒等式的例子．本文深入探研此公式的功能，通过对它的恒等变形，挖掘它在组合数数列求和方面的应用，从而使我们对组合数公式的...     

二类有趣的组合恒等式

二类有趣的组合恒等式湖南岳阳师专肖振纲在拙文［１］中，我们证明了如下的（即原文引理２）引理数列ａ０，ａ１，…，ａｎ（ｎ≥２）成等差数列的充分必要条件为：对任意整数ｉ，当１≤ｉ≤ｎ—１时，恒有并由此给出了二项式定理的一个推广．本文继续在此引理的基础上，...     

二项式定理在数列求和中的应用

所谓二项式定理,是指下面这个公式：（ａ＋ｂ）ｎ＝Ｃ０ｎａｎ＋Ｃ１ｎａｎ－１ｂ＋…＋Ｃｒｎａｎ－ｒｂｒ＋…＋Ｃｎｎｂｎ．其中ｎ是自然数．本文将举例说明,对于某些含有组合数的数列求和问题,充分利用二项式定理,可以使问题得到巧妙的解决．一、系数成等差数列的...     

二项式定理及二项式系数的性质

二项式定理及二项式系数的性质湖南省浏阳市第九中学谢世裕［基本概念］１·二项式定理是在学生熟悉乘法公式和组合数性质的基础上提出的重要定理．这个公式即为二项式定理，右边的多项式叫做（ａ十ｂ）ｎ的二项展开式，其中的系数叫做二项式系数．２．（ａ＋ｂ）ｎ的二项...     

关于Motzkin－Schroder数列

关于Ｍｏｔｚｋｉｎ－Ｓｃｈｒｏｄｅｒ数列乐茂华，陈锡庚（湛江师范学院数学系湛江５２４０４８）（茂名教育学院数学系茂名５２５０００）关键词：递推数列；生成函数；组合恒等式ＡＭＳ（１９９１）主题分类：１１Ｂ３７．对于正整数ｋ，设Ｓ一［（ｋ＋ｌ）／２〕．一...     

一类与多项式相关的组合恒等式

一类与多项式相关的组合恒等式王良成（四川省达县师专６３５０００）本文给出一类与多项式相关的组合恒等式，由此可以产生许多有用的组合恒等式．定理设是１＋１次多项式，则证明１°当ｎ－１，即ｆ（ｘ）＝ａｘ２＋ｂｘ＋ｃ０时，则即（１）式成立．２°假设ｎ＝ｋ，即...     

构造组合数模型巧证组合恒等式

证明组合恒等式,一般是利用组合数的性质、数学归纳法、二项式定理等,通过一些适当的计算或化简来完成.但是,很多组合恒等式,也可直接利用组合数的意义来证明.即构造一个组合问题的模型,把等式两边看成同一组合问题的两种计算方法,由结论的唯一性,即可证明组合恒等式.例1证明:C     

高考专题复习系列讲座——排列、组合和概率

[考试内容和考试要求]1．排列、组合、二项式定理考试内容：分类计数原理与分步计数原理，排列，排列数公式，组合．组合数公式，组合数的两个性质．二项式定理。二项展开式的性质．     

三角数列恒等式的证明

１引言有关三角数列和、积恒等式的证明问题,在现行高中数学教材中是用数学归纳法证明的,当然数学归纳法是证明这类恒等式的基本方法,但是过程比较繁琐．本文给出一种简洁的证明方法,它可以作为证明数列恒等式的通法．２两个定理定理１如果ｆ（ｎ）－ｆ（ｎ－１）＝ｇ...     

Permanent analogues of determinantal identities related to permutation fixed-points

We present permanent analogues of a determinantal identity due to A. Cayley and a formula computing the determinant of so-called zero-axial matrices, for both the generic commuting and noncommuting cases. The Cayley theorem and its permanental versions are derived using combinatorial interpretation of a classical binomial identity The Theorems concerning zero-axial matrices are gotten by the principle of inclusion-exclusion.     

All binomial identities are orderable

The main result of this paper is to show that all binomial identities are orderable. This is a natural statement in the combinatorial theory of finite sets, which can also be applied in distributed computing to derive new strong bounds on the round complexity of the weak symmetry breaking task.Furthermore, we introduce the notion of a fundamental binomial identity and find an infinite family of values, other than the prime powers, for which no fundamental binomial identity can exist.     

Series transformation formulas of Euler type,Hadamard product of series,and harmonic number identities

The Hadamard multiplication theorem for series is used to establish several Euler-type series transformation formulas. As applications we obtain a number of binomial identities involving harmonic numbers and an identity for the Laguerre polynomials. We also evaluate in a closed form certain power series with harmonic numbers.     

Counting hypercubes in hypercubes

Subcubes of a hypercube are counted in three different ways, yielding a new graph theory interpretation of a known combinatorial identity. From this and the binomial inversion some additional combinatorial identities related to hypercubes are obtained.     

由Abel定理导出的组合恒等式、概率分布及应用

一般的损失分布为伽玛分布、威布尔分布或他们的特例指数分布 .但有些损失分布具有 U形特征 .本文根据二项式定理的另一种拓展形式 ,导出了一类新型组合恒等式 ,并将其推广到多维情形 .由此导出一类新的分布列 .最后给出了该恒等式和分布在图论和质量管理中的应用 .     

Jordan canonical form of Pascal-type matrices via sequences of binomial type

In this paper, we study the Jordan canonical form of the generalized Pascal functional matrix associated with a sequence of binomial type, and demonstrate that the transition matrix between the generalized Pascal functional matrix and its Jordan canonical form is the iteration matrix associated with the binomial sequence. In addition, some combinatorial identities are derived from the corresponding matrix factorization.     

Congruences for Catalan and Motzkin numbers and related sequences

We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic: group actions, induction, and Lucas’ congruence for binomial coefficients come into play. A number of our results settle conjectures of Cloitre and Zumkeller. The Thue-Morse sequence appears in several contexts.     

The Combinatorics of Polynomial Sequences

We present a combinatorial model for the several kinds of polynomial sequences of binomial type and develop many of the theorems about them from this model. In the first section, we present a prefab model for the binomial formula and the generating-function theorem. In Sec. 2, we introduce the notion of U-graph and give examples of binomial prefabs of U-graphs. The umbral composition of U-graphs provides an interpretation of umbral composition of polynomial sequences in Sees. 3 and 5. Rota's interpretation of the Stirling numbers of the first kind as sums of the Mobius function in the partition lattice inspired our model for inverse sequences of binomial type in Sec. 4. Section 6 contains combinatorial proofs of several operator-theoretic results. The actions of shift operators and delta operators are explained in set-theoretic terms. Finally, in Sec. 6 we give a model for cross sequences and Sheffer sequences which is consistent with their decomposition into sequences of binomial type. This provides an interpretation of shift-invariant operators. Of course, all of these interpretations require that the coefficients involved be integer and usually non-negative as well.     

分裂求和技巧与组合恒等式

对上式利用对角线相消法则得(2).证毕.注意到定义(1)可以视为普通升降乘和 q 升阶乘算子的概括和拓广.因此可以期望由部分和公式(2)得到部分关于二项式系数和 Gauss 二项式系数的卷积型组合恒等式.下面便进行这方面的讨论.     

自然数等幂和的一个新的通项递归公式

从几何观点出发,利用二项式定理推导出自然数等幂和的一个新的通项递归公式,比已有的组合逼近式、递推式更简洁有效.