On a partition identity of Lehmer

Euler's identity equates the number of partitions of any non-negative integer n into odd parts and the number of partitions of n into distinct parts. Beck conjectured and Andrews proved the following companion to Euler's identity: the excess of the number of parts in all partitions of n into odd parts over the number of parts in all partitions of n into distinct parts equals the number of partitions of n with exactly one even part (possibly repeated). Beck's original conjecture was followed by generalizations and so-called “Beck-type” companions to other identities.In this paper, we establish a collection of Beck-type companion identities to the following result mentioned by Lehmer at the 1974 International Congress of Mathematicians: the excess of the number of partitions of n with an even number of even parts over the number of partitions of n with an odd number of even parts equals the number of partitions of n into distinct, odd parts. We also establish various generalizations of Lehmer's identity, and prove related Beck-type companion identities. We use both analytic and combinatorial methods in our proofs.     

Partial theta function identities from Wang and Ma's conjecture

Recently, Wang and Ma proposed a conjecture, embedding the Andrews–Warnaar partial theta function identity in an infinite family of such identities. In this paper we use q-series methods to give a proof of the Wang–Ma conjecture. We also present a result which may be regarded as the inverse of the Wang–Ma conjecture.     

Order statistics applications to queueing and scheduling problems

We prove several basic combinatorial identities and use them in two applications: the queue inference engine (QIE) and earliest due date rule (EDD) scheduling. Larson (1990) introduced the QIE. His objective was to deduce the behavior of a multiserver queueing system without observing the queue. With only a Poisson arrival assumption, he analyzed the performance during a busy period. Such a period starts once all servers are busy with the queue empty, and it ends as soon as a server becomes idle. We generalize the standard order statistics result for Poisson processes, and show how to sample a busy period in the M/M/c system. We derive simple expressions for the variance of the total waiting time in the M/M/c and M/D/1 queues given that n Poisson arrivals and departures occur during a busy period. We also perform a probabilistic analysis of the EDD for a one-machine scheduling problem with earliness and tardiness penalties. The schedule is without preemption and with no inserted idle time. The jobs are independent and each may have a different due date. For large n, we show that the variance of the total penalty costs of the EDD is linear in n. The mean of the total penalty costs of the EDD is known to be proportional to the square root of n (see Harel (1993)). This revised version was published online in June 2006 with corrections to the Cover Date.     

实Clifford分析中,广义双正则函数向量的带位移带共轭的非线性边值问题

首先得到了广义正则函数向量的 Plemelj公式 ,然后利用积分方程的方法和 Arzela- Ascoli定理 ,讨论了实 Clifford分析中广义双正则函数向量的带位移带共轭的非线性边值问题解的存在性 .     

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.     

两个Rogers—Ramanujan恒等式的新证法

利用一个基本超几何函数的变换公式及其最基本的求和公式，对Ｇｅｓｓｅｌ Ｉ．和Ｓｔａｎｔｏｎ Ｄ。发现的两个Ｒｏｇｅｒｓ－Ｒａｍａｎｕｊａｎ恒等式，给出一种新的、更为简单的证明。     

General relations and identities for order statistics from non-independent non-identical variables

Some recurrence relations and identities for order statistics are extended to the most general case where the random variables are assumed to be non-independent non-identically distributed. In addition, some new identities are given. The results can be used to reduce the computations considerably and also to establish some interesting combinatorial identities.     

A bibasic hypergeometric transformation associated with combinatorial identities of the Rogers-Ramanujan type

During the last five decades, a number of combinatorial generalizations and interpretations have occurred for the identities of the Rogers-Ramanujan type. The object of this paper is to give a most general known analytic auxiliary functional generalization which can be used to give combinatorial interpretations of generalizedq-identities of the Rogers-Ramanujan type. The derivation realise the theory of basic hypergeometric series with two unconnected bases.     

On identities of the Rogers-Ramanujan type

A generalized Bailey pair, which contains several special cases considered by Bailey (Proc. London Math. Soc. (2), 50, 421–435 (1949)), is derived and used to find a number of new Rogers-Ramanujan type identities. Consideration of associated q-difference equations points to a connection with a mild extension of Gordon’s combinatorial generalization of the Rogers-Ramanujan identities (Amer. J. Math., 83, 393–399 (1961)). This, in turn, allows the formulation of natural combinatorial interpretations of many of the identities in Slater’s list (Proc. London Math. Soc. (2) 54, 147–167 (1952)), as well as the new identities presented here. A list of 26 new double sum–product Rogers-Ramanujan type identities are included as an Appendix. 2000 Mathematics Subject Classification Primary—11B65; Secondary—11P81, 05A19, 39A13     

Clifforrd分析中广义双正则函数的(线性)非线性边值问题

本文研究Ｃｌｉｆｏｒｄ分析中广义双正则函数的一个非线性边值问题：Ａ（ｔ１，ｔ２）Ｗ＋＋（ｔ１，ｔ２）＋Ｂ（ｔ１，ｔ２）Ｗ＋－（ｔ１，ｔ２）＋Ｃ（ｔ１，ｔ２）Ｗ－＋（ｔ１，ｔ２）＋Ｄ（ｔ１，ｔ２）Ｗ－－（ｔ１，ｔ２）＝ｇ（ｔ１，ｔ２）ｆｔ１，ｔ２，Ｗ＋＋（ｔ１，ｔ２），Ｗ＋－（ｔ１，ｔ２），Ｗ－＋（ｔ１，ｔ２），Ｗ－－（ｔ１，ｔ２）［］．先讨论解的积分表示式，再研究几个奇异算子，最后用Ｓｃｈａｕｄｅｒ不动点原理（压缩映射定理）证明了解的存在性（唯一性）．目前还没有见到其它国内外学者研究广义双正则函数的非线性边值问题．本文推广了Ｆ．Ｂｒａｃｋｓ，Ｗ．Ｐｉｎｃｋｅｔ［１０］，ＬｅＨｕａｎｇ Ｓｏｎ［１１］，Ｒ．Ｐ．ＧｉｌｂｅｒｔａｎｄＪ．Ｌ．Ｂｕｃｈｎａｎ［１５］和黄沙［１３］的工作