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...     

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.     

加权Herz型Hardy空间上的Littlewood-Paley g_λ~*函数

讨论了在q=2的情形下,Littlewood-Paley gλ*函数在加权Herz型Hardy空间中的有界性,即当0相似文献   

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-Shaped liquid crystal trimers with dual terminal cholesteryl moieties: synthesis and concomitant of N*, SmA and cholesteric glassy phases

Two homologous series of λ-shaped chiral liquid crystal trimers composed of a laterally substituted benzylidene-aniline as the central core armed by two cholestryl ester moieties via odd-even alkyl spacer are synthesised. All the compounds are mesogenic exhibiting both chiral nematic (N*) and SmA phases except for trimers bearing long spacers (n = 9–10). A pronounced odd-even effect is observed on the phase transition temperatures and clearing enthalpies when the spacer length is varied in which the even-parity members show higher value. The widening of N* phase upon elongation of the alkyl spacer can be interpreted as a result of the destabilisation of SmA phase. Full recrystallisation from the cholesteric phase upon cooling is not observed for all the compounds bearing long spacer, especially oxydecanoyl spacer. Instead, the anisotropic fluid vitrified to form the cholesteric glassy state characteristic of polymer at low temperature. This finding is also evident from the polarising optical microscope whereby non-crystalline texture which resembles the Grandjean texture with various reflection colours is observed upon cooling to low temperature. The effect of the lateral substituents on the liquid-crystalline behaviour is also discussed.     

Relations between some basic results derived from two kinds of topologies for a random locally convex module

The purpose of this paper is to exhibit the relations between some basic results derived from the two kinds of topologies (namely the (ε,λ)-topology and the stronger locally L⁰-convex topology) for a random locally convex module. First, we give an extremely simple proof of the known Hahn-Banach extension theorem for L⁰-linear functions as well as its continuous variant. Then we give the relations between the hyperplane separation theorems in [D. Filipovi?, M. Kupper, N. Vogelpoth, Separation and duality in locally L⁰-convex modules, J. Funct. Anal. 256 (2009) 3996-4029] and a basic strict separation theorem in [T.X. Guo, H.X. Xiao, X.X. Chen, A basic strict separation theorem in random locally convex modules, Nonlinear Anal. 71 (2009) 3794-3804]: in the process we also obtain a very useful fact that a random locally convex module with the countable concatenation property must have the same completeness under the two topologies. As applications of the fact, we prove that most of the previously established principal results of random conjugate spaces of random normed modules under the (ε,λ)-topology are still valid under the locally L⁰-convex topology, which considerably enriches financial applications of random normed modules.