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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Appell Polynomials and Asymptotic Expansions

Some properties of the Appell polynomials are studied and analyzed. Various formulas and expressions for the Appell quotient are derived and connection with asymptotic expansions is presented.     

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.     

First crossing time,overshoot and Appell–Hessenberg type functions

We consider a general insurance risk model with extended flexibility under which claims arrive according to a point process with independent increments, their amounts may have any joint distribution and the premium income is accumulated following any non-decreasing, possibly discontinuous, real valued function. Point processes with independent increments are in general non-stationary, allowing for an arbitrary (possibly discontinuous) claim arrival cumulative intensity function which is appealing for insurance applications. Under these general assumptions, we derive a closed form expression for the joint distribution of the time to ruin and the deficit at ruin, which is remarkable, since as we show, it involves a new interesting class of what we call Appell–Hessenberg type functions. The latter are shown to coincide with the classical Appell polynomials in the Poisson case and to yield a new class of the so called Appell–Hessenberg factorial polynomials in the case of negative binomial claim arrivals. Corollaries of our main result generalize previous ruin formulas e.g. those obtained for the case of stationary Poisson claim arrivals.     

Biorthogonal decompositions of the Radon transform

Summary. The concept of singular value decompositions is a valuable tool in the examination of ill-posed inverse problems such as the inversion of the Radon transform. A singular value decomposition depends on the determination of suitable orthogonal systems of eigenfunctions of the operators , . In this paper we consider a new approach which generalizes this concept. By application of biorthogonal instead of orthogonal functions we are able to apply a larger class of function sets in order to account for the structure of the eigenfunction spaces. Although it is preferable to use eigenfunctions it is still possible to consider biorthogonal function systems which are not eigenfunctions of the operator. With respect to the Radon transform for functions with support in the unit ball we apply the system of Appell polynomials which is a natural generalization of the univariate system of Gegenbauer (ultraspherical) polynomials to the multivariate case. The corresponding biorthogonal decompositions show some advantages in comparison with the known singular value decompositions. Vice versa by application of our decompositions we are able to prove new properties of the Appell polynomials. Received October 19, 1993     

Some characterizations of appell andℊ-appell polynomials

Summary Several characterizations are given for the wellknown Appell polynomials and for their basic analogues: the -Appell polynomials defined by Equation (3.3)below. The main results contained in Theorems 1, 2and 3of the present paper, and the applications considered in Section 2,are believed to be new. Some interesting connections with earlier results are also indicated.Supported, in part, by NSERC (Canada) grant A-7353.     

Some binomial formulas of the generalized Appell form

The authors aim here at finding all the generalizations of the binomial formula that are given by a generating-function of the generalized Appell form for a sequence of Newton polynomials. The formulas obtained include the well-known q-analogue of the binomial formula, several formulas involving hyperbolic functions, a trigonometric analogue, and some formulas involving the geometric and the exponential series.     

Monogenic pseudo‐complex power functions and their applications

The use of a non‐commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of different types of polynomials. In this sense, the present paper investigates the intrinsic relationship between two different types of monogenic Appell polynomials. Several authors payed attention to the construction of complete sets of monogenic Appell polynomials, orthogonal with respect to a certain inner product, and used them advantageously for the study of problems in 3D‐elasticity and other problems. Our goal is to show that, as consequence of the binomial nature of those generalized Appell polynomials, their inner structure is determined by interesting combinatorial relations in which the central binomial coefficients play a special role. As a byproduct of own interest, a Riordan–Sofo type binomial identity is also proved. Copyright © 2013 John Wiley & Sons, Ltd.     

Polynomials,random walks and risk processes: a multivariate framework

This paper is concerned with two families of multivariate polynomials: the Appell polynomials and the Abel-Gontcharoff polynomials. Both families are well-known in the univariate case, but their multivariate version is much less standard. We first provide a simple interpretation of these polynomials through particular constrained random walks on a lattice. We then derive nice analytical results for two special cases where the parameters of the polynomials are randomized. Thanks to the interpretation and randomization of the polynomials, we can derive new results and give other insights for the study of two different risk problems: the ruin probability in a multiline insurance model and the size distribution in a multigroup epidemic.     

Determinant Forms for Extended Heat Functions and Expansions of Solutions of Associated Cauchy Problems

We briefly review series solutions of differential equations problems of the second order that lead to coefficients expressed in terms of determinants. Derivative type formulas involving a generating function with several parameters are developed for these determinant coefficients in first order problems. These permit constructing determinant forms for the heat polynomials and their Appell transforms. Hadamard's theorem for bounding determinants and conical regions are used to deduce simplified versions of expansion theorems involving these polynomials and associated Appell transforms. Extended versions of the heat equation are also considered.     

Hermite-based Appell polynomials: Properties and applications

By employing certain operational methods, the authors introduce Hermite-based Appell polynomials. Some properties of Hermite-Appell polynomials are considered, which proved to be useful for the derivation of identities involving these polynomials. The possibility of extending this technique to introduce Hermite-based Sheffer polynomials (for example, Hermite-Laguerre and Hermite-Sister Celine's polynomials) is also investigated.     

Une identité pour des polynômes d'Appell

In this paper, we establish an identity for some Appell polynomials generalizing explicit formulas for generalized Bernoulli numbers and polynomials.     

On Paravector Valued Homogeneous Monogenic Polynomials with Binomial Expansion

The aim of this note is to study a set of paravector valued homogeneous monogenic polynomials that can be used for a construction of sequences of generalized Appell polynomials in the context of Clifford analysis. Therefore, we admit a general form of the vector part of the first degree polynomial in the Appell sequence. This approach is different from the one presented in recent papers on this subject. We show that in the case of paravector valued polynomials of three real variables, there exist essentially two different types of such polynomials together with two other trivial types of polynomials. The proof indicates a way of obtaining analogous results in the case of polynomials of more than three variables.