Applications of an Explicit Formula for the Generalized Euler Numbers

The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.     

Euler数和高阶Euler数的推广

The purpose of this paper is to define the generalized Euler numbers and the generalized Euler numbers of higher order, their recursion formula and some properties were established, accordingly Euler numbers and Euler numbers of higher order were extended.     

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.     

Note on calculating wiener numbers of molecular graphs with symmetry

The authors provided a simple method for calculating Wiener numbers of moleculargraphs with symmetry in 1997. This paper intends to further improve on it and simplifies thecalculation of the Wiener numbers of the molecular graphs.     

递推数列的非线性表达式的构造

A type of nonlinear expressions of Lucas sequences are established inspired by Hsu [A nonlinear expression for Fibonacci numbers and its consequences.J.Math.Res.Appl.,2012,32(6):654–658].Using the relationships between the Lucas sequence and other linear recurring sequences satisfying the same recurrence relation of order 2,i.e.,the Horadam sequences,we may transfer the identities of Lucas sequences to the latter.     

Curvature, Diameter and Bounded Betti Numbers

In this paper, we introduce the notion of bounded Betti numbers, and show that the bounded Betti numbers of a closed Riemannian n-manifold (M, g) with Ric (M) ≥ -(n - 1) and Diam (M) ≤ D are bounded by a number depending on D and n. We also show that there are only finitely many isometric isomorphism types of bounded cohomology groups (H*(M), || · ||∞) among closed Riemannian manifold (M, g) with K(M) ≥-1 and Diam (M) ≤ D.     

SOME CLASSES OF DIFFERENCE SEQUENCES OF FUZZY NUMBERS DEFINED BY A SEQUENCE OF MODULI

In this article, we introduce and examine some properties of new difference sequence spaces of fuzzy numbers defined using a sequence of modulus functions.     

Unified characteristic numbers and solutions of equations for birth and death processes with barriers

The state 0 of a birth and death process with state space E = {0, 1, 2,....} is a barrier which can be classified into four kinds： reflection, absorption, leaping reflection, quasi-leaping reflection. For the first, second and fourth barriers, the characteristic numbers of different forms have been introduced. In this paper unified characteristic numbers for birth and death processes with barriers were introduced, the related equations were solved and the solutions were expressed by unified characteristic numbers. This paper concerns work solving probability construction problem of birth and death processes with leaping reflection barrier and quasi-leaping reflection barrier.     

On Weighted Possibilistic Mean,Variance and Correlation of Interval-valued Fuzzy Numbers

In this paper, the concept of weighted possibilistic mean of interval- valued fuzzy number is first introduced. Further, the notions of weighted possibilistic variance, covariance and correlation of interval-valued fuzzy numbers are presented. Meantime, some important properties of them and relationships between them are studied.     

关于高阶Genocchi数的一些恒等式

讨论了高阶Genocchi数的性质,建立了一些包含高阶Genocchi数和高阶Euler-Bernoulli数的恒等式.     

A generalization of the formula for the triangular number of the sum and product of natural numbers

This note generalizes the formula for the triangular number of the sum and product of two natural numbers to similar results for the triangular number of the sum and product of r natural numbers. The formula is applied to derive formula for the sum of an odd and an even number of consecutive triangular numbers.     

Introduction to Fibonacci and Lucas hybrinomials

ABSTRACT

The hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper, we introduce and study the Fibonacci and Lucas hybrinomials, i.e. polynomials, which are a generalization of the Fibonacci hybrid numbers and the Lucas hybrid numbers, respectively.     

Sums of products of Cauchy numbers

In this paper, we consider a kind of sums involving Cauchy numbers, which have not been studied in the literature. By means of the method of coefficients, we give some properties of the sums. We further derive some recurrence relations and establish a series of identities involving the sums, Stirling numbers, generalized Bernoulli numbers, generalized Euler numbers, Lah numbers, and harmonic numbers. In particular, we generalize some relations between two kinds of Cauchy numbers and some identities for Cauchy numbers and Stirling numbers.     

2500年研究探寻相亲数

设σ(n)为n的所有正因子(包括1和n本身在内)之和.正整数对(m,n)被称之为相亲数(或双亲数,因为这种数总是成双成对出现的)如果他们满足 σ(m)=σ(n) = m + n.如果n=n, σ(m)=2m,则m被称之为完全数(或单亲数,因为这种数总是单独出现的).更一般的,如果κ个(κ＞2)正整数(m1,m2,…mmk)满足下列条件σ(m1)=m1+m2,σ(m2)=m2+m3,σ(mk)=mκ+m1.则这κ个正整数被称之为多亲数.第一对相亲数(220,284)是在2500年前的古希腊数学家毕达哥拉斯发现的.不过迄今为止,人们对相亲数的情况、尤其对相亲数的分布情况仍然知之甚少.与相亲数有关的难题、尤其是悬而未决千百年的难题还很多就是在今夭,我们仍然不知道是不是有无穷多对相亲数,我们甚至连一个生成相亲数的充分必要条件(定义除外)都没有.在这篇文章中,我们试图给出人类在2500年的漫长历史长河中研究、探寻相亲数的大致情况与重要结果,并着重介绍从古至今生成相亲数的各种数值方法与代数方法.完全数的研究探寻史几乎与相亲数的研究探寻史是一样长的.比如2350年前的古希腊数学家欧几理德就在其数学名著<几何原本>中列出了前四个完全数,不过迄今为止,人们总共也只找到39个完全数,并且这些完全数还都是偶完全数.至于有没有奇完全数的存在,则是一个悬而未决两千多年的著名数学难题.最早的两串多亲数(一串为5个.另一串为28个),则是由法国数学家Poulet于1918年发现的.多亲数的研究探寻史虽然比相亲数的研究探寻史要短得多,但目前人们对它们的注意力与日俱增.由于相亲数与完全数及多亲数密切相关、紧密相连(我们可以将其统一称之为亲和数,因为它们都与相关数的因子和有关),因此在本文中,我们除了要讨论介绍相亲数外,也将顺便介绍完全数与多亲数的研究与探寻简史、以及人们在研究探寻这些数时所获得的一些重要结果.附注截止2004年3月25日作者校勘清样时,人们已经发现了共40个完全数和6262871对相亲数.     

关于模糊数贴近度问题的研究

利用模糊集和区间数的贴近度理论,讨论了模糊数的贴近度问题.通过区间数与模糊数之间的关系,根据区间数贴近度的一般表示形式,给出了构造一些模糊数贴近度的具体计算公式的方法;并通过实例说明了所得到的贴近度公式的有效性和实用性,解决了常用贴近度公式所不能解决的问题.     

New convolutions for complete and elementary symmetric functions

In this paper, we give new relationships between complete and elementary symmetric functions. These results can be used to discover and prove some identities involving r-Whitney numbers, Jacobi–Stirling numbers, Bernoulli numbers and other numbers that are specializations of complete and elementary symmetric functions.     

高阶Bernoulli数与Stirling数的几个恒等式

利用第一、二类高阶Bernoulli数和二类Stirling数S1(n,k),S2(n,k)的定义.研究了二类高阶Bernoulli数母函数的幂级数展开,揭示了二类高阶Bernoulli数之间以及与第一类Stirling数S1(n,k)、第二类Stirling数S2(n,k)之间的内在联系,得到了几个关于二类高阶Bernoulli数和第一类Stirling数S1(n,k)、第二类Stirling数S2(n,k)之间有趣的恒等式.     

Peters type polynomials and numbers and their generating functions: Approach with p‐adic integral method

The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p‐adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well‐known special numbers and polynomials are presented. By using p‐adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol‐type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol‐Bernoulli polynomials, the Apostol‐Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well‐known formulas. Finally, two open problems for interpolation functions for Apostol‐type Peters numbers and polynomials are revealed.