Spaces of vector-valued Dirichlet series in a half plane

Dirichlet series with real frequencies which represent entire functions on the complex plane C have been investigated by many authors. Several properties such as topological structures, linear continuous functionals, and bases have been considered. Le Hai Khoi derived some results with Dirichlet series having negative real frequencies which represent holomorphic functions in a half plane. In the present paper, we have obtained some properties of holomorphic Dirichlet series having positive exponents, whose coefficients belong to a Banach algebra.     

A Class of Generalized Hausdorff Means

In the research of divergent series the important idea is that choose a suitable trasformation have to be of regularity, of sequence of partial sum of the series. The Hausdorff transformation is H=δμδ in matrix form, in which μ is any diagonal transformation and δ is so-called δ-transformation. In this paper, we construct a class of generalized Housdorff transformation using the Gould-Hsu inverse series, and give necessary and sufficient condition of that it shou ld be regular.     

拉格朗日展开定理的两个基本应用

The present paper offers two likely neglected applications of the classical Lagrange expansion formula.One is a unified approach to some age-old derivative identities originally due to Pfaff and Cauchy.Another is two explicit matrix inversions which may serve as common generalizations of some known inverse series relations.     

OSCILLATION CONVERGENT THEOREMS FOR THE V^2-INTEGRAL

The V^t-integral as defined in[2], which is eqnivalent to M^2-integrsl as defined in Trigonometre series by Zygmund is used to sum trigonometric seies in[1]. In this paper, some convergent theorems of V^2-integral are established.     

ASYMPTOTIC PROPERTIES OF THE MOMENT CONVERGENCE FOR NA SEQUENCES

It is well-known that the complete convergence theorem for i.i.d. random variables has been an active topic since the famous work done by Hsu and Robbins [6]. Chow [4]obtained a moment version of Hsu and Robbins series. However, the series tends to infinity whenever ∈ goes to zero, so it is of interest to investigate the asymptotic behavior of the series as ∈ goes to zero. This note gives some limit theorems of the series generated by moments for NA random variables.     

MULTIFOLD ANALOGUES OF GOULD-HSU INVERSIONS AND THEIR APPLICATIONS

By means of difference calculus, a multifold analog of Gould-Hsu inversion is established. Asconcrete examples, some multifold inverse relations are presented. By developing a standard prooftechnique for combinatorial identities, we demonstrate Mohanty-Hanna's convolution formulas almostmechanically. In addition, an application of reciprocal formulas to interpolation process is alsosketched.     

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.     

ON GENERALIZED COMPLETE (p,q)-ELLIPTIC INTEGRALS

In this paper,we study the generalized complete(p,q)-elliptic integrals of the first and second kind as an application of generalized trigonometric functions with two parameters,and establish the monotonicity,generalized convexity and concavity of these functions.In particular,some Turán type inequalities are given.Finally,we also show some new series representations of these functions by applying Alzer and Richard's methods.     

The gerber-shiu expected discounted penalty function for Lévy insurance risk processes

In this paper,we study a general Lévy risk process with positive and negative jumps.A renewal equation and an infinite series expression are obtained for the expected discounted penalty function of this risk model.We also examine some asymptotic behaviors for the ruin probability as the initial capital tends to infinity.     

Value distribution theory and the research of Yang Lo

Value distribution theory is concerned with the position and frequency of solutions of the equation f(z) = a. Here f may be entire, i.e. an everywhere convergent power series or meromorphic, i.e. the ratio of two such series, or a function in some other domains, such as an angle or a disk. Yang Lo’s significant contributions to this area will be highlighted. Some of his important contributions to normal families will also be described.     

Multifold analogues of Gould-Hsu inversions and their applications

By means of difference calculus, a multifold analog of Gould-Hsu inversion is established. As concrete examples, some multifold inverse relations are presented. By developing a standard proof technique for combinatorial identities, we demonstrate Mohanty-Hanna's convolution formulas almost mechanically. In addition, an application of reciprocal formulas to interpolation process is also sketched.     

一些反演关系的反演链

本文主要提出和讨论了一般反演关系的反演链和反演对的理论.在此基础上,具体给出Gould-Hsu反演的两类反演链和Gould-Hsu反演的q-模拟的相关问题.最后,本文结合一些常用的反演对,建立了众多的组合恒等式.     

Closed form expressions for some series over certain trigonometric integrals

The series (3) and (4), where T(x) denotes trigonometric integrals (2), are represented as series in terms of Riemann zeta and related functions using the sums of the series (5) and (6), whose terms involve one trigonometric function. These series can be brought in closed form in some cases, where closed form means that the series are represented by finite sums of certain integrals. By specifying the function φ(y) appearing in trigonometric integrals (2) we obtain new series for some special types of functions as well as known results.     

On <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-integrability of a special double sine series formed by its blocks

In this paper we deal with a special double sine trigonometric series formed by its blocks. This type of trigonometric series is of particular interest since its blocks always are bounded, that is, under some additional assumptions the sum-function of such series always exists. We give some conditions under which such sum-function is integrable of power p ∈ {2, 3,... }, as well as is integrable with some natural weight.     

数项级数的收敛与重排

讨论了级数重排问题,在条件收敛级数与其重排级数的项之间增加一定的约束,可以得到原级数与重排级数收敛到同一数值的结论.     

Multiple harmonic series related to multiple Euler numbers

In this paper, we define the multiple Euler numbers and consider some multiple harmonic series of Mordell-Tornheim's type, which is a partial sum of the Mordell-Tornheim zeta series defined by Matsumoto. Indeed, we prove a certain reducibility of these series as well as the multiple zeta values.     

Concerning an infinite series of Ramanujan related to the natural logarithm

We consider an infinite series, due to Ramanujan, which converges to a simple expression involving the natural logarithm. We show that Ramanujan’s series represents a completely monotone function, and explore some of its consequences, including a non-trivial family of inequalities satisfied by the natural logarithm, some formulas for the Euler–Mascheroni constant, and a recurrence satisfied by the Bernoulli numbers. We also provide a one-parameter generalization of Ramanujan’s series, which includes as a special case another related infinite series evaluation due to Ramanujan.     

Limits of areas under lattice paths

We consider sequences of polynomials which count lattice paths by area. In some cases the reversed polynomials approach a formal power series as the length of the paths tend to infinity. We find the limiting series for generalized Schröder, Motzkin, and Catalan paths. The limiting series for Schröder paths and their generalizations are shown to count partitions with restrictions on the multiplicities of odd parts and no restrictions on even parts. The limiting series for generalized Motzkin and Catalan paths are shown to count generalized Frobenius partitions and some related arrays.     

Neumann series and Lommel functions of two variables

We obtain several new results for Neumann series of Bessel functions as well as for its various special cases. The generalization of some well-known results for these kind of series, such as the Graf's addition theorem, are also established.     

Some evaluation of harmonic number sums

In this paper, by using the method of partial fraction decomposition and integral representations of series, we establish some expressions of series involving harmonic numbers and binomial coefficients in terms of zeta values and harmonic numbers. Furthermore, we can obtain some closed form representations of sums of products of quadratic (or cubic) harmonic numbers and reciprocal binomial coefficients, and some explicit evaluations are given as applications. The given representations are new.