与Hermite多项式和广义Laguerre多项式相关联的双正交多项式系统

生成函数刻画了正交多项式的很多重要性质.本文的主要目的是根据生成函数的特点研究正交多项式类之间的渐近关系.本文拓展了Lee及其合作者的工作,构造一类双正交多项式系统,并由此构造出分别渐近于Hermite多项式和广义Laguerre多项的函数列;给出渐近于Hermite多项式和广义Laguerre多项的函数列的判定定理.作为这些性质的应用,可以直接获得若干正交多项式和组合多项式的渐近表示,从而验证了揭示超几何多项式渐近关系的Askey格式成立.     

B样条在一些渐近组合问题中的应用

本文考察了B样条函数及其导数的渐近性质,并给出了收敛阶;考察了经典Eulerian数和两类广义Eulerian数的渐近性质;给出了以Hermite多项式表示的细化Eulerian数的渐近形式.Carlitz等人利用中心极限定理得到Eulerian数渐近公式的逼近阶为43阶.利用样条方法,我们得到更为精确的逼近阶.将样条方法引入到组合数的渐近分析中,为离散对象的研究提供了一种新的分析方法.     

广义n阶Euler-Bernoulli多项式

本文得到了广义n阶Euler数和广义n阶Bernoulli数,广义n阶Euler多项式和广义n阶Bernoulli多项式的关系式。     

Genocchi积分多项式及其性质

本文研究了Genocchi积分多项式的性质.利用生成函数的方法,得到了Genocchi积分多项式的一些组合恒等式,揭示了Genocchi积分多项式和Genocchi多项式、Bernoulli多项式、Genocchi数、Bernoulli数、Euler数之间的关系.     

关于Lucas多项式平方和的恒等式

利用广义Lucas多项式L n(x,y)的性质,通过构造组合和式T n(x,y;tx2),结合Bernoulli多项式的生成函数和Euler多项式的生成函数,采用分析学中的方法,得到两个有关L2n(x,y)的恒等式.并从这一结果出发,得到了两个推论,推广了相关文献的一些结果.     

高阶Bernoulli多项式和高阶Euler多项式的新计算公式

使用发生函数方法,利用两种第一类Stirling数给出高阶Bernoulli多项式和高阶Euler多项式的简捷计算公式.     

一种广义周期Besov类的多元周期多项式样条逼近

本文证明多元多项式周期样条空间是某些多元周期光滑函数类的关于Kolmogorov n-宽度的弱渐近极子空间．给出了广义周期Besov类的一种推广，得到了空间元素的一种表示定理，不仅给出了一种多元周期多项式样条算子．而且证明了所得的结果．     

递归序列与高阶项式

引　　言关于递归序列与Euler-Bernoulli数和多项式、递归序列与高阶Euler-Bernoulli数和多项式的关系问题的研究一直是国内外许多学者感兴趣的课题,并有了许多研究成果(见[1]～[7]).本文首先对Euler-Bernoulli数和多项式、高阶Euler-Bernoulli数和多项式进行推广,提出高阶多元Euler数和多项式、高阶多元Bernoulli数和多项式的定义,然后讨论它们与递归序列的关系,文中得出的结果是P.F.Byrd[1],R.P.Kelisky[2]和Zhangzhizheng[3]的相应结果的推广和深化.2　定义和引理定义2.1　k阶s元Euler数E(k)v1…vs和k阶s元Bernoulli数B(k)v1…v…     

Bernoulli多项式与Euler—Maclaurin公式的推广

本文提出了广义Bernoulli多项式与广义Bernoulli数,并借此得到了一类含两端点连续阶导数值求积公式的误差渐近式和推广的Euler-Maclaurin求和公式.借助于计算机代数系统进行了公式的机械推导,并列出了部分推导结果.     

高阶多元Norlund Euler-Bernoulli多项式

给出了高阶多元Nrlund Euler多项式和高阶多元Nrlund Bernoulli多项式的定义,讨论了它们的一些重要性质,建立了一些包含递归序列和上述多项式的恒等式.     

New characterization of Appell polynomials

We prove characterizations of Appell polynomials by means of symmetric property. For these polynomials, we establish a simple linear expression in terms of Bernoulli and Euler polynomials. As applications, we give interesting examples. In addition, from our study, we obtain Fourier expansions of Appell polynomials. This result recovers Fourier expansions known for Bernoulli and Euler polynomials and obtains the Fourier expansions for higher order Bernoulli–Euler's one.     

The relation of the d-orthogonal polynomials to the Appell polynomials

We are dealing with the concept of d-dimensional orthogonal (abbreviated d-orthogonal) polynomials, that is to say polynomials verifying one standard recurrence relation of order d + 1. Among the d-orthogonal polynomials one singles out the natural generalizations of certain classical orthogonal polynomials. In particular, we are concerned, in the present paper, with the solution of the following problem (P): Find all polynomial sequences which are at the same time Appell polynomials and d-orthogonal. The resulting polynomials are a natural extension of the Hermite polynomials.

A sequence of these polynomials is obtained. All the elements of its (d + 1)-order recurrence are explicitly determined. A generating function, a (d + 1)-order differential equation satisfied by each polynomial and a characterization of this sequence through a vectorial functional equation are also given. Among such polynomials one singles out the d-symmetrical ones (Definition 1.7) which are the d-orthogonal polynomials analogous to the Hermite classical ones. When d = 1 (ordinary orthogonality), we meet again the classical orthogonal polynomials of Hermite.     

A determinantal approach to Sheffer–Appell polynomials via monomiality principle

In this article, the Sheffer and Appell polynomials are combined to introduce the family of Sheffer–Appell polynomials by using operational methods. The determinantal definition and other properties of the Sheffer–Appell polynomials are established. As particular cases of these polynomials, the Sheffer–Bernoulli and Sheffer–Euler polynomials are introduced and their determinantal definitions are obtained. The operational correspondence between the Appell and Sheffer–Appell polynomials is used to derive the results for the Sheffer–Appell polynomials. Certain results for the Hermite–Appell and Laguerre–Appell polynomials are also obtained.     

Srivastava–Pintér theorems for 2D‐Appell polynomials and their applications

Recently, Srivastava and Pintér proved addition theorems for the generalized Bernoulli and Euler polynomials. Luo and Srivastava obtained the anologous results for the generalized Apostol–Bernoulli polynomials and the generalized Apostol–Euler polynomials. Finally, Tremblay et al. gave analogues of the Srivastava–Pintér addition theorem for general family of Bernoulli polynomials. In this paper, we obtain Srivastava–Pintér type theorems for 2D‐Appell Polynomials. We also give the representation of 2D‐Appell Polynomials in terms of the Stirling numbers of the second kind and 1D‐Appell polynomials. Furthermore, we introduce the unified 2D‐Apostol polynomials. In particular, we obtain some relations between that family of polynomials and the generalized Hurwitz–Lerch zeta function as well as the Gauss hypergeometric function. Finally, we present some applications of Srivastava–Pintér type theorems for 2D‐Appell Polynomials. Copyright © 2013 John Wiley & Sons, Ltd.     

Addition theorems for the Appell polynomials and the associated classes of polynomial expansions

Various interesting and potentially useful properties and relationships involving the Bernoulli, Euler and Genocchi polynomials have been investigated in the literature rather extensively. Recently, the present authors (Srivastava and Pinter in Appl Math Lett 17:375–380, 2004) obtained addition theorems and other relationships involving the generalized Bernoulli polynomials ${B_n^{(\alpha)}(x)}$ and the generalized Euler polynomials ${E_n^{(\alpha)}(x)}$ of order α and degree n in x. The main purpose of this sequel to some of the aforecited investigations is to give several addition formulas for a general class of Appell sequences. The addition formulas, which are derived in this paper, involve not only the generalized Bernoulli polynomials ${B_n^{(\alpha)}(x)}$ and the generalized Euler polynomials ${E_n^{(\alpha)}(x)}$ , but also the generalized Genocchi polynomials ${G_n^{(\alpha)}(x)}$ , the Srivastava polynomials ${\mathcal{S}_{n}^{N}\left( x\right)}$ , several general families of hypergeometric polynomials and such orthogonal polynomials as the Jacobi, Laguerre and Hermite polynomials. Some umbral-calculus generalizations of the addition formulas are also investigated.     

Moment systems and orthogonal polynomials in several variables

The first part of this paper deals with general moment (“Appell”) systems on $\text{R}$^N generated by a Hamiltonian function H(x, D) and also with representations of GL(N) on the associated spaces of polynomials. The second part discusses the theory of Bernoulli generators on $\text{R}$^N determining systems of orthogonal polynomials that are extensions of the Meixner polynomials to several variables. Linear actions for these spaces are discussed. Some tensors related to the general Bernoulli generators are considered.     

Resurgence relation and global asymptotic analysis of orthogonal polynomials via the Riemann-Hilbert approach

A global asymptotic analysis of orthogonal polynomials via the Riemann-Hilbert approach is presented,with respect to the polynomial degree.The domains of uniformity are described in certain phase variables.A resurgence relation within the sequence of Riemann-Hilbert problems is observed in the procedure of derivation.Global asymptotic approximations are obtained in terms of the Airy function.The system of Hermite polynomials is used as an illustration.     

Reciprocal relations of Bernoulli and Euler numbers/polynomials

By means of the symmetric summation theorem on polynomial differences due to Chu and Magli [Summation formulae on reciprocal sequences. European J Combin. 2007;28(3):921–930], we examine Bernoulli and Euler polynomials of higher order. Several reciprocal relations on Bernoulli and Euler numbers and polynomials are established, including some recent ones obtained by Agoh Shortened recurrence relations for generalized Bernoulli numbers and polynomials. J Number Theory. 2017;176:149–173.