Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind

Recently, the authors introduced some generalizations of the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials (see [Q.-M. Luo, H.M. Srivastava, J. Math. Anal. Appl. 308 (2005) 290-302] and [Q.-M. Luo, Taiwanese J. Math. 10 (2006) 917-925]). The main object of this paper is to investigate an analogous generalization of the Genocchi polynomials of higher order, that is, the so-called Apostol-Genocchi polynomials of higher order. For these generalized Apostol-Genocchi polynomials, we establish several elementary properties, provide some explicit relationships with the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials, and derive various explicit series representations in terms of the Gaussian hypergeometric function and the Hurwitz (or generalized) zeta function. We also deduce their special cases and applications which are shown here to lead to the corresponding results for the Genocchi and Euler polynomials of higher order. By introducing an analogue of the Stirling numbers of the second kind, that is, the so-called λ-Stirling numbers of the second kind, we derive some basic properties and formulas and consider some interesting applications to the family of the Apostol type polynomials. Furthermore, we also correct an error in a previous paper [Q.-M. Luo, H.M. Srivastava, Comput. Math. Appl. 51 (2006) 631-642] and pose two open problems on the subject of our investigation.     

窄线SeyfertⅠ型星系

本文将概述窄线SeyfertⅠ型星系 (包括具有类似性质但光度较大的类星体 )的主要观测特征、物理内涵及其理论解释和在活动星系核内部结构上的反映。窄线SeyfertⅠ型星系 (NLS1 ,以后NLS1都认为包括NLSI星系和NLS1类星体 )具有窄的允许线Hβ,它兼有Sy1和Sy2的某些特征。它具有强的FeⅡ光学线和弱的 [OⅢ ]λ5 0 0 7禁线 ,这明显不同于Sy2。在观测相关量的主成分统计分析中表明NLS1是活动星系核第一主向量表现为极端的一类。这包括 :( 1 )HβFWHM取极小 ;( 2 )FeⅡλ45 70 /Hβ 取极大 ;( 3)SⅢ ]λ1 892 /CⅢ ]λ1 90 9取极大 ,亦一定范围内宽线区电子密度取大值 ;( 4 )CⅣλ1 5 4 9线中心蓝移取极大 ;( 5 )高光度NLS1的软X射线光子数谱指数Γx 取极大 ;( 6)X射线相对流量变化方差最大等。基于以上观测结构和相应统计规律 (观测量与HβFWHM的关系 ) ,解释NLS1现象最佳的模型是高相对吸积率L/LEdd模型。NLS1比之典型宽线活动星系核具有较低的中心黑洞质量MBH。进一步 ,还发现这一模型是活动星系核主...     

Efficient Synthesis of Some New 2,3-Dimethylquinoxaline-1-oxides Using Bis(acetoxy)phenyl-λ 3-iodane as an Oxidant

A series of 2-(arylimino)-3-(hydroxyimino)butanes 3a–g, easily accessed by the condensation of variously substituted anilines 1a–g with biacetyl monoxime 2, were efficiently cyclized to afford 2,3-dimethylquinoxaline-1-oxides 4a–g using bis(acetoxy)phenyl-λ ³-iodane as an oxidant. This methodology utilizes a commercially available and environmental benign oxidant to achieve the title compounds in excellent yields under mild conditions.

[Supplementary materials are available for this article. Go to the publisher's online edition of Synthetic Communications® for the following free supplemental resource(s): Full experimental and spectral details.]     

(λ,μ)-模糊子环和理想的进一步讨论

在(λ,μ)-模糊子群的基础上,进一步刻画了(λ,μ)-模糊子环和理想.详细讨论了(λ,μ)-模糊子环和理想的性质,以及它们与截集的关系,给出了相应结论.     

Rearrangements in Spiro-λ4-sulfane Chemistry

The reactions of possible spiro-λ⁴-sulfane precursor diaryl sulfoxides with dehydrating agents have been studied resulting in the formation of cyclic anhydride, spiro-λ⁴-sulfane, cyclic sulfonium-ylide, lactam-sulfoxide, cyclic sulfonimide and cyclic sulfilimine type of intermediates or products. The structures of the intermediates or products depend on the functional groups attached to the aryl rings at ortho positions and on the reaction conditions applied.     

STUDIES ON ORGANOPHOSPHORUS COMPOUNDS XXX. MASS SPECTROSCOPIC INVESTIGATION OF 2-ALKYL-2-OXO-1,3,2-DIOXA-PHOSPHORINANE AND-PHOSPHEPANE

Abstract

The mass spectra of 2-alkyl-2-oxo-1,3,2-dioxa-phosphorinane and-phosphepane showed that the ring opening was in competition with the cleavage of the P[sbnd]C bond. According to the fragmentation pathway, which was dependent on the structure of exocyclic substituents on phosphorus, the 2-alkyl-2-oxo-1,3,2-di-oxa-phosphorinanes can be classified in two categories. The main process in category A was the ring opening and/or C[sbnd]C bond cleavage. While in category B the cleavage of P[sbnd]C bond was predominant. However, for 2-alkyl-2-oxo-1,3,2-dioxa-phosphepane. no matter how the structure of 2-alkyl group was, the ring opening was a dominant process.     

λ噬菌体溶源/裂解转换调控与定态熵

被λ噬菌体感染的大肠杆菌是研究生物基因调控网络的重要模板之一. 根据该调控系统操纵子与调控蛋白相互作用特点, 利用热力学配分函数描述调控蛋白与操纵子的结合概率, 在给出该调控系统的动力学分岔的基础上, 计算系统各定态的热力学熵. 结果显示, 溶源态和裂解态较鞍点和分岔点具有更低的熵值, 且溶源态的熵最低, 说明处于溶源态的系统具有更高的生物序.     

Convergence of compressible Euler-Poisson system to incompressible Euler equations

In this paper, we study the quasi-neutral limit of compressible Euler-Poisson equations in plasma physics in the torus T^d. For well prepared initial data the convergence of solutions of compressible Euler-Poisson equations to the solutions of incompressible Euler equations is justified rigorously by an elaborate energy methods based on studies on an λ-weighted Lyapunov-type functional. One main ingredient of establishing uniformly a priori estimates with respect to λ is to use the curl-div decomposition of the gradient.     

双菱体λ/4消色差相位延迟器的设计

根据菲涅尔全内反射相变理论,给出了双菱体λ/4消色差器的结构设计、性能分析和测量方法.由有效通光孔径和光线追迹设计出BK7玻璃在波长532 nm时相位延迟λ/4的双菱体的结构,用作532 nm至1 064 nm波长范围的标准λ/4相位延迟器.理论分析了入射角变化和波长变化对双菱体相位延迟的影响,当入射角变化限制在±4.3°以内时,其影响得到补偿；波长从532 nm到1 064 nm产生的误差为－0.65°.采用椭偏法中的消光技术,分别实测了双菱体在532 nm和1 064 nm波长下的相位延迟为：90.08±0.14°和88.99±0.1°,可知两不同波长产生的相位延迟误差为－1.09°.