Explicit Formulae for Values of Dedekind Zeta Functions of Two Kinds of Cyclotomic Fields

Let K = Q(m) denote the m-th cyclotomic field, and K+ its maximal real subfield, where m =exp is an m-th primary root of unity. Let K (s) denote the Dedekind zeta function ofK. For prime integers m = p, Fumio Hazama recently in [1] obtained formulae for calculating special values of K(s) and K+(s), i.e., calculating formulae of K+(1 - n) and for positive integers n, which are the newest results of a series of his work in many years (see [1-3]).Here we develop Hazama's work for prime integ…     

Gould-Hsu反演链及其应用

In this paper,by means of Gould-Hsu inverse series relations,we establish several Gould-Hsu inversion chains.As consequence,some new transformation formulae as well as some famous hypergeometric series identities are derived.     

On Finite Forms of Certain Bilateral Basic Hypergeometric Series and Their Applications

In this paper we derive finite forms of the summation formulas for bilateral basic hypergeometric series 3ψ3,4ψ4 and 5ψ5.We therefrom obtain the summation formulae obtained recently by Wenchang CHU and Xiaoxia WANG.As applications of these summation formulae,we deduce the well-known Jacobi's two and four square theorems,a formula for the number of representations of an integer n as sum of four triangular numbers and some theta function identities.     

ON QUADRATURE FORMULAE FOR SINGULAR INTEGRALS OF ARBITRARY ORDER

Some quadrature formulae for the numerical evaluation of singular integrals of arbitrary order are established and both the estimate of remainder and the convergence of each quadrature formula derived here are also given.     

CONVERGENCE AND GROWTH OF MULTIPLE DIRICHLET SERIES

This article investigates the convergence and growth of multiple Dirichlet series. The Valiron formula of Dirichlet series is extended to n-tuple Dirichlet series and an equivalence relation between the order of n-tuple Dirichlet series and its coefficients and exponents is obtained.     

The First Power Mean of the Inversion of L-Functions Weighted by Quadratic Gauss Sums

The main purpose of this paper is to use estimates for character sums and analytic methods to study the first power mean of the inversion of Dirichlet L-functions with the weight of general quadratic Gauss sums, and three asymptotic formulae are obtained.     

Hybrid Mean Value of the Hyper Cochrane Sums

The main purpose of this paper is to use the analytic methods to study the hybrid mean value involving the hyper Cochrane sums, and give several sharp asymptotic formulae.     

关于一个普遍本源公式及其应用

Here presented is a further investigation on a general source formula(GSF) that has been proved capable of deducing more than 30 special formulas for series expansions and summations in the author's recent paper [On a pair of operator series expansions implying a variety of summation formulas.Anal.Theory Appl.,2015,31(3):260–282].It is shown that the pair of series transformation formulas found and utilized by He,Hsu and Shiue [cf.Disc.Math.,2008,308:3427–3440] is also deducible from the GSF as consequences.Thus it is found that the GSF actually implies more than 50 special series expansions and summation formulas.Finally,several expository remarks relating to the(Σ?D) formula class are given in the closing section.     

与中心数相关的一类级数求和公式

This paper provides a pair of summation formulas for a kind of combinatorial series involvingak＋b m as a factor of the summand. The construction of formulas is based on a certain series transformation formula [2, 7, 9] and by making use of the C-numbers [3]. Various consequences and examples including several remarkable classic identities are presented to illustrate some applications of the formulas obtained.     

关于错排数与Bell数的一些循环公式

In this paper, a further investigation for the number of Derangements and Bell numbers is performed, and some new recursion formulae for the number of Derangements and Bell numbers are established by applying the generating function methods and Padé approximation techniques. Illustrative special cases of the main results are also presented.     

交错Mordell-Tornheim和Witten多重级数

本文通过对几个交错多重调和级数的讨论,统一地得到了交错Mordell-Tornheim和Witten多重级数的一些已有的和新的表达式.     

一般随机缺项三角级数表示断片的Bouligand维数

该文对［１］中的Ｂｏｕｌｉｇａｎｄ维数计算公式进行了改进，用对称原理和简化原理，得到了一般随机缺项三角级数所表示断片的Ｂｏｕｌｉｇａｎｄ维数的一些计算公式．     

Further summation formulae related to generalized harmonic numbers

By employing the univariate series expansion of classical hypergeometric series formulae, Shen [L.-C. Shen, Remarks on some integrals and series involving the Stirling numbers and ζ(n), Trans. Amer. Math. Soc. 347 (1995) 1391-1399] and Choi and Srivastava [J. Choi, H.M. Srivastava, Certain classes of infinite series, Monatsh. Math. 127 (1999) 15-25; J. Choi, H.M. Srivastava, Explicit evaluation of Euler and related sums, Ramanujan J. 10 (2005) 51-70] investigated the evaluation of infinite series related to generalized harmonic numbers. More summation formulae have systematically been derived by Chu [W. Chu, Hypergeometric series and the Riemann Zeta function, Acta Arith. 82 (1997) 103-118], who developed fully this approach to the multivariate case. The present paper will explore the hypergeometric series method further and establish numerous summation formulae expressing infinite series related to generalized harmonic numbers in terms of the Riemann Zeta function ζ(m) with m=5,6,7, including several known ones as examples.     

Abel’s lemma on summation by parts and partial <Emphasis Type="Italic">q</Emphasis>-series transformations

The partial sums of basic hypergeometric series are investigated by means of the modified Abel lemma on summation by parts. Several transformation and summation formulae for well-poised, quadratic, cubic and quartic q-series are established. This work was partially supported by National Natural Science Foundation for the Youth (Grant No. 10801026)     

Extensions of the well-poised and elliptic well-poised Bailey lemma

We establish a number of extensions of the well-poised Bailey lemma and elliptic well-poised Bailey lemma. As application we prove some new transformation formulae for basic and elliptic hyper-geometric series, and embed some recent identities of Andrews, Berkovich and Spiridonov in a well-poised Bailey tree.     

Convergence acceleration of orthogonal series

The aim of this paper is to verify efficiency of two acceleration methods for orthogonal series (more strictly, for series defined at the beginning of Section 1). These methods are quite different although they use the same transform of such a series given there. The first method (Section 3) has some features common with Levin’s and Weniger’s methods. It may be profitably used in numerical calculations for a vast class of series. The second one (Sections 4 and 5) is somewhat similar to the Euler–Knopp transform of power series. Also this method is numerically realizable but more important is that for a narrower class of series, including some ones having applications in physics, it gives explicit analytic formulae of their transform.      

Minton-Karlsson identities and summation formulae involving generalized harmonic numbers

By applying the derivative operator to Minton and Karlsson's hypergeometric series identities, several interesting summation formulae involving generalized harmonic numbers are established.     

Abel's lemma on summation by parts and basic hypergeometric series

Basic hypergeometric series identities are revisited systematically by means of Abel's lemma on summation by parts. Several new formulae and transformations are also established. The author is convinced that Abel's lemma on summation by parts is a natural choice in dealing with basic hypergeometric series.     

一些超几何级数恒等式的新的证明

The Abel＇s lemma on summation by parts is employed to evaluate terminating hypergeometric series. Several summation formulae are reviewed and some new identities are established.