组合恒等式

本文利用函数１ｆ（ｘ）展开式定理,导出一批新的组合恒等式．     

Divided Differences and Combinatorial Identities

We present an algebraic theory of divided differences which includes confluent differences, interpolation formulas, Liebniz's rule, the chain rule, and Lagrange inversion. Our approach uses only basic linear algebra. We also show that the general results about divided differences yield interesting combinatorial identities when we consider some suitable particular cases. For example, the chain rule gives us generalizations of the identity used by Good in his famous proof of Dyson's conjecture. We also obtain identities involving binomial coefficients, Stirling numbers, Gaussian coefficients, and harmonic numbers.     

Vertex Algebras and Combinatorial Identities

In the 1980's, J. Lepowsky and R. Wilson gave a Lie-theoretic interpretation of Rogers–Ramanujan identities in terms of level 3 representations of affine Lie algebra sl(2,C)~. When applied to other representations and affine Lie algebras, Lepowsky and Wilson's approach yielded a series of other combinatorial identities of the Rogers–Ramanujan type. At about the same time, R. Baxter rediscovered Rogers–Ramanujan identities within the context of statistical mechanics. The work of R. Baxter initiated another line of research which yielded numerous combinatorial and analytic generalizations of Rogers–Ramanujan identities. In this note, we describe some ideas and results related to Lepowsky and Wilson's approach and indicate the connections with some results in combinatorics and statistical physics.     

Concerning Various Combinatorial Identities

According to H. W. Gould [3] the notation S: p/q indicates a kind of binomial coefficient summation in which p denote the number of binomial coefficients appearing in The numerator term of the summation, and q has the same meaning for the denominator term. In order to classify combinatorial identities (binomial coefficient summations) more precisely, we may introduce the concept "degree" for a combinatorial identity of the type S: p/q, say. We may say that an identity for the sum     

Identities for Combinatorial Extremal Theory

Let be the set {1, 2, ..., n}, and let Ø be the emptyset. Let G be the family of all non-empty sets of subsets of. For A G and X , put An important discovery is the following. 1991 Mathematics SubjectClassification 05D05.     

Skew-Symmetric Differential Operatorsand Combinatorial Identities

 In this paper, we classify all one-function skew-symmetric and super skew-symmetric differential operators by four combinatorial identities related to hyperbolic functions. Received 17 June 1997; in revised form 14 October 1997     

Partitions with Non-Repeating Odd Parts and Combinatorial Identities

Continuing our earlier work on partitions with non-repeating odd parts and q-hypergeometric identities, we now study these partitions combinatorially by representing them in terms of 2-modular Ferrers graphs. This yields certain weighted partition identities with free parameters. By special choices of these parameters, we connect them to the Göllnitz-Gordon partitions, and combinatorially prove a modular identity and some parity results. As a consequence, we derive a shifted partition theorem mod 32 of Andrews. Finally we discuss basis partitions in connection with the 2-modular representation of partitions with non-repeating odd parts, and deduce two new parity results involving partial theta series.     

Combinatorial Interpretations for Identities Using Chromatic Partitions

Czechoslovak Mathematical Journal - We provide combinatorial interpretations for three new classes of partitions, the so-called chromatic partitions. Using only combinatorial arguments, we show...     

新型组合恒等式（一）

新型组合恒等式是研讨别开生面的几类组合孪生恒等式组的问题．本要主要研讨互逆类的组合孪生恒等式组，该类可分为多项式型、二项式定理型、指数函数型以及三角函数（或双曲函数）型等四型，一批成双出现的新结果。与许多著名数列（Ｆｉｂｏｎａｃｃｉ数列、Ｂｅｒｎｏｕｌｌｉ数、Ｅｕｌｅｒ数以及二项式定理系数数列等）有着密切关系．此外，本人还研讨了一类特殊行列式的性质及其应用。     

Identities on Algebras and Combinatorial Properties of Binary Words

Doklady Mathematics - Polynomial identities and codimension growth of nonassociative algebras over a field of characteristic zero are considered. A new approach is proposed for constructing...     

Explicit Formulas and Combinatorial Identities for Generalized Stirling Numbers

In this paper, a modified approach to the multiparameter non-central Stirling numbers via differential operators, introduced by El-Desouky, and new explicit formulae of both kinds of these numbers are given. Also, some relations between these numbers and the generalized Hermite and Truesdel polynomials are obtained. Moreover, we investigate some new results for the generalized Stirling-type pair of Hsu and Shiue. Furthermore some interesting special cases, new combinatorial identities and a matrix representation are deduced.     

Combinatorial Proofs of Various q-Pell Identities via Tilings

Recently, Benjamin, Plott, and Sellers proved a variety of identities involving sums of Pell numbers combinatorially by interpreting both sides of a given identity as enumerators of certain sets of tilings using white squares, black squares, and gray dominoes. In this article, we state and prove q-analogues of several Pell identities via weighted tilings.     

Application of Generalized Legendre Polynomial in Combinatorial Identities

ＡｐｐｌｉｃａｔｉｏｎｏｆＧｅｎｅｒａｌｉｚｅｄＬｅｇｅｎｄｒｅＰｏｌｙｎｏｍｉａｌｉｎＣｏｍｂｉｎａｔｏｒｉａｌＩｄｅｎｔｉｔｉｅｓＺｈａｎｇＺｈｉｚｈｅｎｇ（张之正）ＬｅｉＺｈｉｊｕｎ；（雷治军）（ＬｕｏｙａｎｇＴｅａｃｈｅｒｓＣａｌｌｅｇｅ，４...     

由一级数引出的若干新的组合恒等式

基于一个简单级数的三种求和方式,导出了两个重要组合恒等式.并在此基础上,或利用特殊化法、或利用极限方法,又导出了几个新的组合恒等式.     

用组合方法证明Gaussian系数恒等式

设F_q是q个元素的有限域,其中q是素数的幂,F_q~n是F_q上n维向量空间,用[n/m]_q表示Gaussian系数,它可看作为F_q~n的m维子空间的个数.用组合方法证明了几个Gaussian系数恒等式.     

与格路有关的组合恒等式

通过研究格路径的性质得到一类组合恒等式的通式,代入不同的参数给出已有的一些组合恒等式新的简洁证明,并得到一些新的组合恒等式.最后推广得到多项式系数的恒等式.     

Combinatorial Properties of Rogers-Ramanujan Type Identities Arising from Hall-Littlewood Polynomials

Here we consider the q-series coming from the Hall-Littlewood polynomials,

$$\begin{array}{l}{R_{v} (a, b; q) = {\sum_{\mathop {\lambda}\limits_{\lambda_1 \leq a}}} q^{c | \lambda |} P_{2\lambda}(1, q, q^{2}, \ldots ; q^{2b+d}).}\end{array}$$

These series were defined by Griffin, Ono, and Warnaar in their work on the framework of the Rogers-Ramanujan identities. We devise a recursive method for computing the coefficients of these series when they arise within the Rogers-Ramanujan framework. Furthermore, we study the congruence properties of certain quotients and products of these series, generalizing the famous Ramanujan congruence

$$p(5n+4) \equiv 0\quad ({\rm mod}\, 5).$$

     

Pascal算子矩阵在组合恒等式中的应用

定义了四种Pascal算子矩阵,给出了它们的代数性质及它们之间的关系,并且利用二项式型多项式序列、算子及哑运算得到许多组合恒等式.     

Pascal算子矩阵在组合恒等式中的应用(英文)

定义了四种Pascal算子矩阵,给出了它们的代数性质及它们之间的关系,并且利用二项式型多项式序列、算子及哑运算得到许多组合恒等式.