关于正整数的有序分拆的Inplace恒等式的一些注记

本文给出了一类特殊的称之为Inplace有序分拆的两个递推关系式的组合证明. 同时, 我们也得到了关于Inplace 1-2 有序分拆,回文的有序分拆的一些新的恒等式.     

关于Bell多项式及其它的一些恒等式

本文引入了一个新的多项式,即Bell多项式.利用初等数论及组合方法,证明了包含该多项式的一些恒等式.作为这些恒等式的应用,给出了关于Bell数的同余式.     

关于Bell多项式及其它的一些恒等式（英文）

本文引入了一个新的多项式,即Bell多项式.利用初等数论及组合方法,证明了包含该多项式的一些恒等式.作为这些恒等式的应用,给出了关于Bell数的同余式.     

几个新组合恒等式及其应用

得到四个新的组合恒等式,由此给出文献[1]中的一个结论的初等证明.     

一类多项式恒等式及其推广

首先利用母函数法和多项式理论证明了一类多项式恒等式,接着通过求导和积分的方式得到一些新的多项式恒等式,最后给出这些恒等式的特殊形式及其应用.     

一个新的非线性Picone恒等式及其应用（英文）

本文建立一个新的非线性Picone恒等式,它包括一些已有的Picone恒等式.利用这个新的Picone恒等式,我们给出了带奇异项p-Laplace方程的Sturm比较原理,p-Laplace方程组的Liouville定理和带权Hardy不等式.由这里一般的带权Hardy型不等式,我们可以得到几个新的有趣的带权型Hardy不等式.     

一类新的m重Rogers-Ramanujan恒等式及应用

Rogers-Ramanujan恒等式是分拆理论和组合学中著名的恒等式,被广泛的证明和推广.该文应用双边Bailey引理和迭代技巧建立一类新的多重和Rogers-Ramanujan恒等式.     

积分恒等式及其在无穷级数求和中的应用

借助L2[0,π]中标准正交基展开理论,得到积分恒等式,然后运用这个积分恒等式,通过定积分计算给出几个无穷级数和公式的简单证明,同时得到一些新的无穷级数和公式.     

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.     

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     

On a Pair of Identities from Ramanujan's Lost Notebook

Using a pair of two variable series-product identities recorded by Ramanujan in the lost notebook as inspiration, we find some new identities of similar type. Each identity immediately implies an infinite family of Rogers-Ramanujan type identities, some of which are well-known identities from the literature. We also use these identities to derive some general identities for integer partitions.     

q-Pell sequences and two identities of V.A. Lebesgue

We examine a pair of Rogers-Ramanujan type identities of Lebesgue, and give polynomial identities for which the original identities are limiting cases. The polynomial identities turn out to be q-analogs of the Pell sequence. Finally, we provide combinatorial interpretations for the identities.     

PIE-sums: A combinatorial tool for partition theory

PIE-sums are introduced. The method of inclusion-exclusion is applied to a wide range of partition identities which are herein proved for the first time by methods other than generating functions. A general structure is given to these varied identities, and several new examples are presented. The connection between partition identities and Möbius function identities is examined, and the foundations are given for developing this technique.     

Combinatorial identities from contour integrals of rational functions

We employ the theory of contour integrals to systematically investigate three kinds of general combinatorial identities in a unified way. As applications some well-known combinatorial identities are presented as special cases, and several new identities are derived.     

Tenth order mock theta functions in Ramanujan's lost notebook (IV)

Ramanujan's lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously the author proved the first six of Ramanujan's tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan's tenth order mock theta function identities which are expressed by mock theta functions and a definite integral. L. J. Mordell's transformation formula for the definite integral plays a key role in the proofs of these identities. Also, the properties of modular forms are used for the proofs of theta function identities.

     

Identities and linear dependence

Central polynomial identities are used to construct alternating central identities by which new identities are obtained. These identities express the linear dependence ofn ²+1 generic matrices, and so yield slight generalizations and simplified proofs of a result of Formanek, the theorem about Azumaya algebras of M. Artin and a recent result of Cauchon.     

Some formulas related to the Rogers-Ramanujan identities

Summary The paper contains a number of identities related to theRogers-Ramanujan identities. In particular, the formulas (13) and (14) are generalizations of those identities.     

New identities involving generalized Fibonacci and generalized Lucas numbers

This paper presents two new identities involving generalized Fibonacci and generalized Lucas numbers. One of these identities generalize the two well-known identities of Sury and Marques which are recently developed. Some other interesting identities involving the famous numbers of Fibonacci, Lucas, Pell and Pell-Lucas numbers are also deduced as special cases of the two derived identities. Performing some mathematical operations on the introduced identities yield some other new identities involving generalized Fibonacci and generalized Lucas numbers.     

Some combinatorial identities of Sarmanov,Sevast'yanov,and Tarakanov

Sarmanov, Sevast'yanov, and Tarakanov have proved certain combinatorial identities by a combinatorial argument, namely, by counting the number of weighted chains by two different methods. The identities are applied to the summation of infinite series and to the computation of the quartiles of discrete probability distributions. Egorychev and Yuzhakov have proved these and some further identities by applying the multidimensional generalization of the Lagrange expansion. In the present paper the identities are proved in a direct, elementary way.