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．     

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.     

Some Combinatorial and Analytical Identities

We give new proofs and explain the origin of several combinatorial identities of Fu and Lascoux, Dilcher, Prodinger, Uchimura, and Chen and Liu. We use the theory of basic hypergeometric functions, and generalize these identities. We also exploit the theory of polynomial expansions in the Wilson and Askey-Wilson bases to derive new identities which are not in the hierarchy of basic hypergeometric series. We demonstrate that a Lagrange interpolation formula always leads to very-well-poised basic hypergeometric series. As applications we prove that the Watson transformation of a balanced ${_{4}\phi_{3}}$ to a very-well-poised ${_{8}\phi_{7}}$ is equivalent to the Rodrigues-type formula for the Askey-Wilson polynomials. By applying the Leibniz formula for the Askey-Wilson operator we also establish the ${_{8}\phi_{7}}$ summation theorem.     

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     

格点偏序集的一些组合恒等式（英）

This paper is devoted to the counting of j-chains of lattice-point poset. The incidence function and two basic enumerators associated with it are introduced and evaluated. Our results contain a lot of combinatorial sums.     

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。     

Divided Differences and Linearly Recursive Sequences

We show that the theory of divided differences is a natural tool for the study of linearly recurrent sequences. The divided differences functional associated with a monic polynomial w on degree n + 1 yields a vector space isomorphism between the space of polynomials of degree at most equal to n and the space of linearly recurrent sequences f that satisfy the difference equation w(E)f=0 where E is the usual shift operator. Using such isomorphisms, we can translate problems about recurrent sequences into simple problems about polynomials. We present here a new approach to the theory of divided differences, using only generating functions and elementary linear algebra, which clarifies the connections of divided differences with rational functions, polynomial interpolation, residues, and partial fractions decompositions.     

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...