Appell Polynomials Research Articles

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.

Determinantal approach to certain mixed special polynomials related to Gould–Hopper polynomials

In this article, the Gould–Hopper–Sheffer polynomials are introduced by means of generating function and determinantal definition. Certain properties of these polynomials are established and operational relations between the Sheffer and Gould–Hopper–Sheffer polynomials are derived. Examples of some members belonging to the Gould–Hopper–Sheffer and Gould–Hopper–Appell polynomials families are considered and the corresponding numbers for certain polynomials are also obtained.

Bernoulli and Euler Polynomials in Clifford Analysis

In this article, we bring a new set of Bernoulli and Euler special monogenic polynomials. It is based on the umbral calculus adapted to the Clifford setting, which is equivalent to a polynomial representation using a special set of Appell polynomials. Also, we study the convergence properties (the effectiveness) for the Bernoulli and Euler polynomials in Clifford analysis. Finally, some results concerning the mode of increase (the order and type) for these sets are obtained.

Recursion rules for the hypergeometric zeta function

The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a+b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an expression as an infinite product. This provides linear and quadratic recurrences for the hypergeometric zeta function. A family of associated polynomials is characterized as Appell polynomials and the underlying distribution is given explicitly in terms of the zeros of the associated hypergeometric function. These properties are also given a probabilistic interpretation in the framework of beta distributions.

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.

Probabilistic approach to Appell polynomials

In this paper we study Appell polynomials by connecting them to random variables. This probabilistic approach yields, e.g., the mean value property which is fundamental in the sense that many other properties can be derived from it. We also discuss moment representations of Appell polynomials. In the latter part of the paper the presented general theory is applied to study some classical explicitly given polynomials.

On the integral of the product of the Appell polynomials

We establish an explicit formula for the integral of the product of several Appell polynomials.

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 certain generalized q-Appell polynomial expansions

AbstractWe study q-analogues of three Appell polynomials, the H-polynomials, the Apostol-Bernoulli and Apostol-Euler polynomials, whereby two new q-difference operators and the NOVA q-addition play key roles. The definitions of the new polynomials are by the generating function; like in our book, two forms, NWA and JHC are always given together with tables, symmetry relations and recurrence formulas. It is shown that the complementary argument theorems can be extended to the new polynomials as well as to some related polynomials. In order to find a certain formula, we introduce a q-logarithm. We conclude with a brief discussion of multiple q-Appell polynomials.

Some families of differential equations associated with the Hermite-based Appell polynomials and other classes of Hermite-based polynomials

Recently, Khan et al. [S. Khan, G. Yasmin, R. Khan and N. A. M. Hassan, Hermite-based Appell polynomials: Properties and Applications, J. Math. Anal. Appl. 351 (2009), 756-764] defined the Hermite-based Appell polynomials by G(x, y, z, t) := A(t)?exp(xt + yt2 + zt3) = ??,n=0 HAn(x, y, z) tn/n! and investigated their many interesting properties and characteristics by using operational techniques combined with the principle of monomiality. Here, in this paper, we find the differential, integro-differential and partial differential equations for the Hermite-based Appell polynomials via the factorization method. Furthermore, we derive the corresponding equations for the Hermite-based Bernoulli polynomials and the Hermite-based Euler polynomials. We also indicate how to deduce the corresponding results for the Hermite-based Genocchi polynomials from those involving the Hermite-based Euler polynomials.

Linearization of the products of the generalized Lauricella polynomials and the multivariate Laguerre polynomials via their integral representations

In this paper, the authors investigate the linearization problems associated with two families of generalized Lauricella polynomials of the first and second kinds. By means of their multiple integral representations, it is shown how one can linearize the product of two different members of each of these two families of the generalized Lauricella polynomials. Upon suitable specialization of the main results presented in this paper, the corresponding integral representations are deduced for such familiar classes of multivariable hypergeometric polynomials as (for example) the Lauricella polynomials FA(r) in r variables, the Appell polynomials F2 in two variables and the multivariable Laguerre polynomials. Each of these integral representations, which are derived as special cases of the main results in this paper, may also be viewed as a linearization relationship for the product of two different members of the associated family of multivariable hypergeometric polynomials.

A set of finite order differential equations for the Appell polynomials

Let {Rn(x)}n=0∞ denote the set of Appell polynomials which includes, among others, Hermite, Bernoulli, Euler and Genocchi polynomials. In this paper, by introducing the generalized factorization method, for each k∈N, we determine the differential operator {Ln,k(x)}n=0∞ such that Ln,k(x)(Rn(x))=λn,kRn(x), where λn,k=(n+k)!n!−k!. The special case k=1 reduces to the result obtained in [M.X. He, P.E. Ricci, Differential equation of Appell polynomials via the factorization method, J. Comput. Appl. Math. 139 (2002) 231–237]. The differential equations for the Hermite and Bernoulli polynomials are exhibited for the case k=2.

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.

Expansions in Appell polynomials of the convolutions of probability distributions

We use the composition method to analyse the convolutions of probability distributions by emploing the Appell polynomials and Bergström identity. Our approximation is based on the probability distributions which have the inverse generalized measure of bounded variation. The idea to use the accompanying probability distribution eλ(F−E), λ > 0, was first proposed by B.V. Gnedenko [1].

2-Iterated Appell polynomials and related numbers

This article deals with the introduction of Appell type sets of polynomials and associated numbers. The 2-iterated Appell polynomials are introduced and the Bernoulli and Euler based Appell polynomials are deduced as their particular cases. Different sets of polynomials, namely 2-iterated Bernoulli and Euler, Bernoulli–Euler (or Euler–Bernoulli) and their related numbers are considered. The operational relations between these families and Appell polynomials are used on the results of the Bernoulli and Euler polynomials to obtain the results for the corresponding mixed polynomials.

Ruin Probabilities for Risk Models with Ordered Claim Arrivals

Recently, Lefevre and Picard (Insur Math Econ 49:512–519, 2011) revisited a non-standard risk model defined on a fixed time interval [0,t]. The key assumption is that, if n claims occur during [0,t], their arrival times are distributed as the order statistics of n i.i.d. random variables with distribution function F t (s), 0 ≤ s ≤ t. The present paper is concerned with two particular cases of that model, namely when F t (s) is of linear form (as for a (mixed) Poisson process), or of exponential form (as for a linear birth process with immigration or a linear death-counting process). Our main purpose is to obtain, in these cases, an expression for the non-ruin probabilities over [0,t]. This is done by exploiting properties of an underlying family of Appell polynomials. The ultimate non-ruin probabilities are then derived as a limit.

-Pascal and -Wronskian Matrices with Implications to -Appell Polynomials

We introduce a -deformation of the Yang and Youn matrix approach for Appell polynomials. This will lead to a powerful machinery for producing new and old formulas for -Appell polynomials, and in particular for -Bernoulli and -Euler polynomials. Furthermore, the --polynomial, anticipated by Ward, can be expressed as a sum of products of -Bernoulli and -Euler polynomials. The pseudo -Appell polynomials, which are first presented in this paper, enable multiple -analogues of the Yang and Youn formulas. The generalized -Pascal functional matrix, the -Wronskian vector of a function, and the vector of -Appell polynomials together with the -deformed matrix multiplication from the authors recent article are the main ingredients in the process. Beyond these results, we give a characterization of -Appell numbers, improving on Al-Salam 1967. Finally, we find a -difference equation for the -Appell polynomial of degree .

General-Appell Polynomials within the Context of Monomiality Principle

A general class of the 2-variable polynomials is considered, and its properties are derived. Further, these polynomials are used to introduce the 2-variable general-Appell polynomials (2VgAP). The generating function for the 2VgAP is derived, and a correspondence between these polynomials and the Appell polynomials is established. The differential equation, recurrence relations, and other properties for the 2VgAP are obtained within the context of the monomiality principle. This paper is the first attempt in the direction of introducing a new family of special polynomials, which includes many other new special polynomial families as its particular cases.

Linearization of the products of the Carlitz–Srivastava polynomials of the first and second kinds via their integral representations

In this paper, the authors investigate two families of generalized Lauricella polynomials which are known as the Carlitz–Srivastava polynomials of the first and second kinds. By means of their multiple integral representations, it is shown how one can linearize the product of two different members of each of these two families of the Carlitz–Srivastava polynomials. Upon suitable specialization of the main results presented in this paper, the corresponding integral representations are deduced for such familiar classes of multivariable hypergeometric polynomials as (for example) the Lauricella polynomials FD(r) in r variables and the Appell polynomials F1 in two variables. Each of these integral representations, which are derived as special cases of the main results in this paper, may also be viewed as a linearization relationship for the product of two different members of the associated family of multivariable hypergeometric polynomials.

Δ h -Appell sequences and related interpolation problem

A determinantal form for Δ h -Appell sequences is proposed and general properties are obtained by using elementary linear algebra tools. As particular cases of Δ h -Appell sequences the sequence of Bernoulli polynomials of second kind and the one of Boole polynomials are considered. A general linear interpolation problem, which generalizes the classical interpolation problem on equidistant points, is proposed. The solution of this problem is expressed by a basis of Δ h -Appell polynomials. Numerical examples which justify theoretical results on the interpolation problem are given.