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.     

Multi-variable Hermite matrix polynomials: Properties and applications

In this paper, the multi-variable Hermite matrix polynomials are introduced by algebraic decomposition of exponential operators. Their properties are established using operational methods. The matrix forms of the Chebyshev and truncated polynomials of two variable are also introduced, which are further used to derive certain operational representations and expansion formulae.     

On Appell sequences of polynomials of Bernoulli and Euler type

A construction of new sequences of generalized Bernoulli polynomials of first and second kind is proposed. These sequences share with the classical Bernoulli polynomials many algebraic and number theoretical properties. A class of Euler-type polynomials is also presented.     

Jacobi polynomials and generalized Clifford algebra‐valued Appell sequences

Appell sequences in Clifford analysis are defined as polynomial families on which the Heisenberg algebra acts through a raising and a lowering operator satisfying the canonical Heisenberg relation. Recently, these sequences have gained new interest, as they are connected to the topic of special functions (such as harmonic or monogenic Gegenbauer polynomials) and branching rules for certain irreducible representations of the spin group. In this paper, we will explain how Jacobi polynomials appear quite naturally in the setting of Appell sequences related to certain branching problems. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Some properties of Hermite-based Sheffer polynomials

In this paper, the concepts and the formalism associated with monomiality principle and Sheffer sequences are used to introduce family of Hermite-based Sheffer polynomials. Some properties of Hermite-Sheffer polynomials are considered. Further, an operational formalism providing a correspondence between Sheffer and Hermite-Sheffer polynomials is developed. Furthermore, this correspondence is used to derive several new identities and results for members of Hermite-Sheffer family.     

On the multidimensional zeta functions associated with theta functions,and the multidimensional Appell polynomials

We define and study the multidimensional Appell polynomials associated with theta functions. For the trivial theta functions, we obtain the various well-known Appell polynomials. Many other interesting examples are given. To push our study, by Mellin transform, we introduce and investigate the multidimensional zeta functions associated with thetas functions and prove that the multidimensional Appell polynomials are special values at the nonpositive integers of these zeta functions. Using zeta functions techniques, among others, we prove an induction formula for multidimensional Appell polynomials. The last part of this paper is devoted to spectral zeta functions and its generalization associated with Laplacians on compact Riemannian manifolds. From this generalization, we construct new Appell polynomials associated with Riemannan manifolds of finite dimensions.     

On a new family related to truncated exponential and Sheffer polynomials

In this article, the truncated exponential and Sheffer polynomials are combined to introduce the 2-variable truncated-exponential based Sheffer polynomials (2VTESP) by using operational methods. Examples of certain special polynomials belonging to this family are considered. Operational correspondence between the 2VTESP and Sheffer polynomials is established, which is applied to derive the results for some members belonging to the 2VTESP family.     

Convergence of Sums of Appell Polynomials with Infinite Variance

We investigate the convergence of distributions of partial sums of Appell polynomials of a long-memory moving average process X _t with i.i.d. innovations _s in the case where the variance , and the distribution of #x03BE; ₀ ^m belongs to the domain of attraction of an -stable law with 1<< 2. We prove that the limit distribution of partial sums of Appell polynomials is either an -stable Lévy process, or an mth order Hermite process, or the sum of two mutually independent processes depending on the values of , m, and d, where 0X _t.     

Appell systems on Lie groups

Using the probabilistic interpretation of Appell polynomials as systems of moments, we show how to define them in the noncommutative case. The method is based on certain infinite-dimensional representations of local Lie groups. For processes, limit theorems play an essential role in the construction. Polynomial matrix representations of convolution semigroups are a principal feature.     

Appell pseudopolynomials and Erlang-type risk models

Appell polynomials are known to play a key role in certain first-crossing problems. The present paper considers a rather general insurance risk model where the claim interarrival times are independent and exponentially distributed with different parameters, the successive claim amounts may be dependent and the premium income is an arbitrary deterministic function. It is shown that the non-ruin (or survival) probability over a finite horizon may be expressed in terms of a remarkable family of functions, named pseudopolynomials, that generalize the classical Appell polynomials. The presence of that underlying algebraic structure is exploited to provide a closed formula, almost explicit, for the non-ruin probability.     

Approximation properties of the generalized Szasz operators by multiple Appell polynomials via power summability method

In this paper some properties of the generalized Szasz operators by multiple Appell polynomials are given, using into consideration the power summability method. In the first section are given some direct estimation related to the generalized Szasz operators by multiple Appell polynomials, including Korovkin type theorem. In the second section, we give some results related to the weighted spaces of continuous functions and Voronovskaya type theorem. In the third section, we have proved some results related to the statistical convergence of the generalized Szasz operators by multiple Appell polynomials, using into consideration the A− transformation. At the end of the paper are given some illustrative computational examples which make such summability methods (for example, power series method) more useful and fruitful for applications of functional analysis in approximation theory.     

Around operators not increasing the degree of polynomials

We present a generic operator J defined on the vectorial space of polynomial functions and we address the problem of finding the polynomial sequences that coincide with the (normalized) J-image of themselves. The technique developed assembles different types of operators and initiates with a transposition of the problem to the dual space. We provide examples for a J limited to three terms.     

The Appell interpolation problem

A general linear interpolation problem is considered. We will call it the Appell interpolation problem because the solution can be expressed by a basis of Appell polynomials. Some classical and non-classical examples are also considered. Finally, numerical calculations are given.     

Gauss级数与Appell级数

本文介绍了Ｇａｕｓ级数与Ａｐｐｅｌ级数的概念，并说明了这两类级数之间的渊源关系．     

渐近于Hermite多项式的双正交系统

本文利用渐近于Gauss函数的函数类?,给出渐近于Hermite正交多项式的一类Appell多项式的构造方法,使得该序列与?的n阶导数之间构成了一组双正交系统.利用此结果,本文得到多种正交多项式和组合多项式的渐近性质.特别地,由N阶B样条所生成的Appell多项式序列恰为N阶Bernoulli多项式.从而,Bernoulli多项式与B样条的导函数之间构成了一组双正交系统,且标准化之后的Bernoulli多项式的渐近形式为Hermite多项式.由二项分布所生成的Appell序列为Euler多项式,从而,Euler多项式与二项分布的导函数之间构成一组双正交系统,且标准化之后的Euler多项式渐近于Hermite多项式.本文给出Appell序列的生成函数满足的尺度方程的充要条件,给出渐近于Hermite多项式的函数列的判定定理.应用该定理,验证广义Buchholz多项式、广义Laguerre多项式和广义Ultraspherical(Gegenbauer)多项式渐近于Hermite多项式的性质,从而验证超几何多项式的Askey格式的成立.     

Orthogonal Appell systems of monogenic functions in the cylinder

The main goal of this paper is to construct Orthogonal Appell systems of polynomial solutions of the Riesz and Moisil‐Théodoresco systems in finite cylinders of ?³. This will be done in the spaces of square integrable functions over ? and ?. Some important properties of the systems are discussed. Copyright © 2011 John Wiley & Sons, Ltd.