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.     

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.     

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.     

A determinantal approach to Appell polynomials

A new definition by means of a determinantal form for Appell (1880) [1] polynomials is given. General properties, some of them new, are proved by using elementary linear algebra tools. Finally classic and non-classic examples are considered and the coefficients, calculated by an ad hoc Mathematica code, for particular sequences of Appell polynomials are given.     

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.     

Persymmetric determinants

The object of this paper is to develop some of the results in the author's joint paper with Dale [2] concerning the derivatives of persymmetric determinants whose elements are Appell functions.Four new double-sum identities are presented which are valid for arbitrary persymmetric determinants. Two of these identities are applied to give direct proofs of two results in [2], A simple formula is given for the derivative of a Turanian of order n with Appell polynomial elements and the result is applied repeatedly to show that its degree is far lower than expected. It is shown that one particular determinant has simple derivatives of all orders and that its degree too is far lower than expected. The formula for the derivative of (first) cofactors is shown to be extensible in a simple manner to the derivatives of second cofactors.     

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.     

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.     

Bivariate second-order linear partial differential equations and orthogonal polynomial solutions

In this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second-order linear partial differential equations, which are admissible potentially self-adjoint and of hypergeometric type. General formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation, using vector matrix notation. Moreover, Rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given. As illustration, these results are applied to a two parameter monic Appell polynomials. Finally, the non-monic case is briefly discussed.     

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     

On solving certain differential equations with variable coefficients

We show how to solve certain types of linear ordinary differential equations with variable coefficients by using Appell polynomials.     

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.     

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.     

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.     

On the inversion of certain moment matrices

An explicit representation of the elements of the inverses of certain patterned matrices involving the moments of nonnegative weight functions is derived in this paper. It is shown that a sequence of monic orthogonal polynomials can be generated from a given weight function in terms of Hankel-type determinants and that the corresponding matrix inverse can be expressed in terms of their associated coefficients and orthogonality factors. This result enables one to obtain an explicit representation of a certain type of approximants which apply to a wide class of positive continuous functions. Convenient expressions for the coefficients of standard classical orthogonal polynomials such as Legendre, Jacobi, Laguerre and Hermite polynomials are also provided. Several examples illustrate the results.     

Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities

An important question in insurance is how to evaluate the probabilities of (non-) ruin of a company over any given horizon of finite length. This paper aims to present some (not all) useful methods that have been proposed so far for computing, or approximating, these probabilities in the case of discrete claim severities. The starting model is the classical compound Poisson risk model with constant premium and independent and identically distributed claim severities. Two generalized versions of the model are then examined. The former incorporates a non-constant premium function and a non-stationary claim process. The latter takes into account a possible interdependence between the successive claim severities. Special attention will be paid to a recursive computational method that enables us to tackle, in a simple and unified way, the different models under consideration. The approach, still relatively little known, relies on the use of remarkable families of polynomials which are of Appell or generalized Appell (Sheffer) types. The case with dependent claim severities will be revisited accordingly.      

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.     

On a homeomorphism between orbit spaces of linear systems and matrix polynomials

The orbit space of controllable systems under system similarity and the orbit space of matrix polynomials with determinant degree equal to the order of the state matrix under right equivalence are proved to be homeomorphic when the quotient compact–open topology is considered in the latter. As a consequence, the variation of the finite and left Wiener–Hopf structures under small perturbations of matrix polynomials with fixed degree for their determinants is described.     

Functional Non-Central and Central Limit Theorems for Bivariate Appell Polynomials

We consider quadratic forms in bivariate Appell polynomials involving strongly dependent time series. Both the spectral density of these time series and the Fourier transform of the kernel of the quadratic forms are regularly varying at the origin and hence may diverge, for example, like a power function. We obtain functional limit theorems for these quadratic forms by extending the recent results on the convergence of their finite-dimensional distributions. Some of these are functional central limit theorems where the limiting process is Brownian motion. Others are functional non-central limit theorems where the limiting processes are typically not Gaussian or, if they are Gaussian, then they are not Brownian motion.