On a Class of Generalized Sampling Functions

In this note, we discuss a class of so-called generalized sampling functions. These functions are defined to be the inverse Fourier transform of a family of piecewise constant functions that are either square integrable or Lebegue integrable on the real number line. They are in fact the generalization of the classic sinc function. Two approaches of constructing the generalized sampling functions are reviewed. Their properties such as cardinality, orthogonality, and decaying properties are discussed. The interactions of those functions and Hilbert transformer are also discussed.     

论一种用到Howard广义斯特灵数的求和公式

It is shown that Howard＇s degenerate weighted Stirling numbers can be used to construct a fruitful summation formula for a class of formal power series involving generalized factorials.This is achieved with the aid of a series-transformation formula due to He,Hsu and Shiue,and several identities involving generalized Stirling numbers and Bell numbers are given to illustrate the application of the formula obtained.     

A new class of generalized close-to-starlike functions defined by the Srivastava-Attiya operator

In this paper, we consider a new class CS*s,b of generalized close-to-starlike functions, which is defined by means of the Srivastava-Attiya operator Js,b involving the Hurwitz-Lerch Zeta function Φ(z, s, a). Basic results such as inclusion relations, coefficient inequalities and other interesting properties of this class are investigated. Relevant connections of some of the results presented here with those that were obtained in earlier investigations are pointed out briefly.     

Linear Volterra Integral Equations as the Limit of Discrete Systems

We consider the multidimensional abstract linear integral equation of Volterra type x（t） （*）∫Rtα（s)x(s)ds=f(t),t∈R,（1）as the limit of discrete Stieltjes-type systems and we prove results on the existence of continuous solutions. The functions x, α and f are Bauach space-valued defined on a compact interval R of R^n Rt is a subinterval of R depending on t ∈ R and (*) f denotes either the Bochner-Lebesgue integral or the Henstock integral. The results presented here generalize those in [1] and are in the spirit of [3]. As a consequence of our approach, it is possible to study the properties of (1) by transferring the properties of the discrete systems, The Henstock integral setting enables us to consider highly oscillating functions.     

A Filtration of Wick Algebra and Its Application to Quantum SDEs

A family of closed snbalgebras, indexed by R(the set of real numbers), of the Wick algebra is constructed. Fundamental properties of tile family are shown including the increasing property and the right-continuity. The notion of adaptedness to the family is defined for quantum stochastic processes in terms of generalized operators. The existence and uniqueness of solutions adapted to the family is established for quantum stochastic differential equations in terms of generalized operators.     

Prop erties for Certain Sub classes of Analytic Functions Asso ciated with Generalized Multiplier Transformation

In this paper, we introduce certain new subclasses of analytic functions defined by generalized multiplier transformation. By using the differential subordination, we study and investigate various inclusion properties of these classes. Also inclusion properties of these classes involving the integral operator are considered.     

Generllzation of Direct Product and Generation of Sets of Orthogonal Functions

Two new kinds of direct product of matrices are defined. Their properties are investigated. Direct products of matrix and set of continuous functions are also defined. Many complete sets of orthogonal functions, such as those sets given by Walsh [2], Paley [3], Chrestenson[4], and Watari [5], may be generated by these newkinds of direct product. Direct products are also applicable to the generation of sets of piecewise orthogonal functions.     

GENERALIZED FRACTIONAL CALCULUS OF THE ALEPH-FUNCTION INVOLVING A GENERAL CLASS OF POLYNOMIALS

The object of this article is to study and develop the generalized fractional calculus operators given by Saigo and Maeda in 1996. We establish generalized fractional calculus formulas involving the product of ■-function, Appell functionF3 and a general class of polynomials. The results obtained provide unification and extension of the results given by Saxena et al. [13], Srivastava and Grag [17], Srivastava et al. [20], and etc. The results are obtained in compact form and are useful in preparing some tables of operators of fractional calculus.On account of the general nature of the Saigo-Maeda operators, ■-function, and a general class of polynomials a large number of new and known results involving Saigo fractional calculus operators and several special functions notablyH-function,I-function, Mittag-Leffler function, generalized Wright hypergeometric function, generalized Bessel-Maitland function follow as special cases of our main findings.     

Properties of Some Families of Meromorphic Multivalent Functions Associated with Generalized Hypergeometric Functions

We introduce and study two subclasses ?_([α_1])(A, B, λ) and ?_([α_1])~+ (A, B, λ) of meromorphic p-valent functions defined by certain linear operator involving the generalized hypergeometric function. The main object is to investigate the various important properties and characteristics of these subclasses of meromorphically multivalent functions. We extend the familiar concept of neighborhoods of analytic functions to these subclasses. We also derive many interesting results for the Hadamard products of functions belonging to the class ?_([α_1])~+(α, β, γ, λ).     

Ma jorization Results and Coefficients Bounds for a Class of Univalent Functions Asso ciated with Operator

In this paper we introduce a new general subclass n,g ∑ a,λ(A, B,α) of univalent func-tions related the known integral operator and differential operator. Some majorization re-sults for n,g ∑ a,λ(A, B, 1) as well as the other functions are given. Furthermore, we find the coefficients bounds on|a2|and|a3|for functions in?n,g ∑ a,λ(A1, B1, A2, B2,α1,α2), which is the bi-univalent functions defined by n,g ∑ a,λ(A, B,α) and subordination. By giving specific values of the parameters of our main results, several(known or new) consequences of main results are also discussed.     

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.     

A q-Analog of Newton’s Series,Stirling Functions and Eulerian Functions

— Recently, Butzer et al. [BH1, BH2, BHS] have studied some classical combinatorial functions such as factorial functions, Stirling numbers and Eulerian numbers of fractional orders. In the present paper we show that much the same is true in the case of the q-analogs. Meanwhile we give some results for the convergence of a q-Newton interpolation series.     

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.     

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.     

Computation methods for combinatorial sums and Euler‐type numbers related to new families of numbers

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well‐known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright © 2016 John Wiley & Sons, Ltd.     

Generalized higher order Bernoulli number pairs and generalized Stirling number pairs

From a delta series f(t) and its compositional inverse g(t), Hsu defined the generalized Stirling number pair . In this paper, we further define from f(t) and g(t) the generalized higher order Bernoulli number pair . Making use of the Bell polynomials, the potential polynomials as well as the Lagrange inversion formula, we give some explicit expressions and recurrences of the generalized higher order Bernoulli numbers, present the relations between the generalized higher order Bernoulli numbers of both kinds and the corresponding generalized Stirling numbers of both kinds, and study the relations between any two generalized higher order Bernoulli numbers. Moreover, we apply the general results to some special number pairs and obtain series of combinatorial identities. It can be found that the introduction of generalized Bernoulli number pair and generalized Stirling number pair provides a unified approach to lots of sequences in mathematics, and as a consequence, many known results are special cases of ours.     

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.     

关于广义Stirling数偶的同余性

本文证明了广义Ｓｔｉｒｌｉｎｇ数偶的一些同余性质，从而回答了文［５］中的一个猜测．这些结果做为特例推广了已知的关于两类Ｓｔｉｒｌｉｎｇ数的同余性质．     

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.     

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.