Stirling数的概率表示和应用

本文证明了第二类Ｓｔｉｒｌｉｎｇ数Ｓ（ｎ，ｋ）和第一类Ｓｔｉｒｌｉｎｇ数ｓ（ｎ，ｋ）都是常见的随机变量和的矩，从而使概率论的方法和技巧，在组合和式计算、恒等式的发现与证明以及渐近估计等方面得到许多新的结果．     

广义Stirling数在组合与统计方面的应用

本文给出了广义 Ｓｔｉｒｌｉｎｇ数 Ｓ（ｎ,ｋ;α,β,γ）的组合解释以及对统计学方面的某些应用,包括Ｃｈａｒａｌａｍｂｉｄｅｓ－Ｋｏｕｔｒａｓ等人的结果为特例．     

涉及Euler数、Bernoulli数和推广的第一类Stirling数的一些恒等式

利用递推关系把文［１］、［２］中的有关结论推广到一般情形，建立起涉及Ｅｕ－ｌｅｒ数、Ｂｅｒｎｏｕｌｉ数和推广的第一类Ｓｔｉｒｌｉｎｇ数的一些恒等式．     

涉及Euler数,Bernoulli数和推广的第一类Stirling数…

利用递推关系把文［１］、［２］中的有关结论推广到一般情形，建立起涉及Ｅｕｌｅｒ数、Ｂｅｒｎｏｕｌｌｉ数和推广的第一类Ｓｔｉｒｌｉｎｇ数的一些恒等式。     

两个涉及第二类Stirling数的求和规则

本文利用第二类Ｓｔｉｒｌｉｎｇ数给出了两个求和规则，从而重新得到了一些古典组合恒等式的统一证明，并由此给出了一些级数求和的渐近估计。     

一类系数依赖于两个参数的齐次递推式之解

本文拓广了文献［１］的研究范围,给出了一类依赖于两个参数的变系数递推式的解的明显表达式．有关结果,使得象Ｌａh数,两类Ｓｔｉｒｌｉｎｇ数以及置换群中的置换个数等多方面的相应定解问题,皆可以直接解出．这对具大数值双指标的递推式的计算,亦或在其理论的研究方面,都有其作用．     

拟正则半群的圈积嵌入与矩阵同余

给出了拟正则半群的圈积嵌入．研究了幂等元集 Ｅ Ｓ 生成的子半群〈 Ｅ Ｓ〉上的矩阵同余可扩张为 Ｓ上的矩阵同余的条件． 特别地,如果〈 Ｅ Ｓ〉上的最小矩阵同余可扩张为 Ｓ上的矩阵同余,那么〈 Ｅ Ｓ〉上的每个矩阵同余可扩张为 Ｓ上的矩阵同余     

正则纯整群带的算子半群和同余网

正则半群Ｓ的同余格（Ｓ）上的算子Ｋ，ｋ，Ｔ和ｔ定义如下：对于ρ∈Ｓ，ρＫ和ρｋ（ρＴ和ρｔ）分别是与ρ有相同核（迹）的最大和最小同余．我们确定了所有正则纯整群带的同余格上由Ｋ，ｋ，Ｔ和ｔ生成的算子半群．并确定了正则纯整群带上任意同余的同余网．     

半伪补MS-代数的滤子同余关系

在半伪补MS-代数上引入余核滤子,正则滤子和p-滤子的概念,研究了半伪补MS-代数上余核滤子,正则滤子和p-滤子的性质及关系,获得了余核p-滤子生成的同余关系的表达式,论证了余核p-滤子同余关系具有同余一致与同余凝聚性质。     

ConcerningWeightedStirling－typePairs

ＣｏｎｃｅｒｎｉｎｇＷｅｉｇｈｔｅｄＳｔｉｒｌｉｎｇ－ｔｙｐｅＰａｉｒｓＬｅｅｔｓｃｈＣ．ＨｓｕＹｕＨｏｎｇｑｕａｎ（ＤａｌｉａｎＵｎｉｖｅｒｓｉｔｙｏｆＴｅｃｈｎｏｌｏｇｒｙ，Ｄａｌｉａｎ１１６０２４）ＣｏｎｃｅｒｎｉｎｇＷｅｉｇｈｔｅｄＳｔｉｒｌ...     

A UNIFIED APPROACH TO A CLASS OF STERLING-TYPE PAIRS

Here presented is a unified approach to a wide class of symmetric Sfirling number pairs,which is determined by four complex parameters and includes as particular cases various previousextensions of Stirling numbers due to Carlicz, Howard, Koutras, Gould-Hopper, respectively.Certain Schlomilch-type formulas and congruence properties will be also exhibited.     

Generalized incomplete poly-Bernoulli and poly-Cauchy numbers

By using the restricted and associated Stirling numbers of the first kind and by generalizing the (unsigned) Stirling numbers of the first kind, we define the generalized incomplete poly-Cauchy numbers by combining the generalized and the incomplete poly-Cauchy numbers, and study their arithmetical and combinatorial properties. We also study the corresponding generalized incomplete poly-Bernoulli numbers.     

Some combinatorial identities via Fibonacci numbers

The Pascal matrix and the Stirling matrices of the first kind and the second kind obtained from the Fibonacci matrix are studied, respectively. Also, we obtain combinatorial identities from the matrix representation of the Pascal matrix, the Stirling matrices of the first kind and the second kind and the Fibonacci matrix.     

On the analytic extension of Stirling numbers of the first kind

We present an analytic extension of the unsigned Stirling numbers of the first kind that is in a certain sense unique in its coincidence with the Stirling polynomials. We examine and compare our extension to previous extensions of (signed) Stirling numbers of the first kind given by Butzer et al. (2007, J. Difference Equ. Appl., 13) and of the unsigned numbers given by Adamchik (1997, J. Comput. Appl. Math., 79). We also see a connection to the Riemann zeta function.     

两类Stirling数的行列式表示

通过计算两个广义的范德蒙(Vandermonde)行列式,得到了第一类无符号Stirling数和第二类Stirling数的一种新的表示方法:用行列式来表示.     

The Legendre–Stirling numbers

The Legendre–Stirling numbers are the coefficients in the integral Lagrangian symmetric powers of the classical Legendre second-order differential expression. In many ways, these numbers mimic the classical Stirling numbers of the second kind which play a similar role in the integral powers of the classical second-order Laguerre differential expression. In a recent paper, Andrews and Littlejohn gave a combinatorial interpretation of the Legendre–Stirling numbers. In this paper, we establish several properties of the Legendre–Stirling numbers; as with the Stirling numbers of the second kind, they have interesting generating functions and recurrence relations. Moreover, there are some surprising and intriguing results relating these numbers to some classical results in algebraic number theory.     

Reciprocity for multirestricted Stirling numbers

Multirestricted Stirling numbers of the second kind count the number of partitions of a given set into a given number of parts, each part being restricted to at most a fixed number of elements. Multirestricted numbers of the first kind are then defined as elements of the matrix inverse to the matrix of corresponding multirestricted numbers of the second kind. The anomalous sign behavior of these latter numbers makes them impervious to combinatorial analysis. In answer to a conjecture that has remained open for several years, we derive a reciprocity law for multirestricted Stirling numbers using algebraic techniques based on polynomial recursions. As corollaries, we obtain new recurrence relations for multirestricted numbers, and a new algebraic derivation of the reciprocity law for Stirling numbers.     

New properties of r-Stirling series

The summation of some series involving the Stirling numbers of the first kind can be found in several works but there is no such a computation for Stirling numbers of the second kind let alone the r-Stirlings. We offer a comprehensive survey and prove new results.      

广义m阶Bernoulli数和广义m阶Euler数的计算公式

使用发生函数方法,利用第一类Stirling数和第二类Stirling数分别给出广义m阶Bernoulli数和广义m阶Euler数的计算公式.     

Plethystic Exponential Polynomials and Plethystic Stirling Numbers

In this article we introduce a plethystic generalization of the exponential polynomials and their umbral inverses. We obtain recursive formulas for both families of polynomials, and use them to get recursions for the plethystic Stirling numbers of the first and second kind and for the plethystic Bell numbers. Finally, we apply Bergeron's S-species to obtain a Dobinsky formula for the plethystic exponential polynomials and a close formula for the plethystic Stirling numbers of the second kind.