Appell Polynomials Research Articles

Recurrence relation for the Appell sequences

In this paper, we present several explicit formulas for Appell polynomials in terms of the weighted Stirling numbers of the second kind. We also provide a unified approach to obtain a three-term recurrence formula for the computation of Appell polynomials. Several examples are given to illustrate our results. As an application, we define and study a new class of polynomials that we call SPI polynomials. They are related to sums of powers of integers.

Approximation by Szász‐Chlodowsky type operators associating 2D‐Appell polynomials

AbstractIn this article, a Chlodowsky type generalization of Szász operators is considered in terms of the 2D Appell polynomials. Convergence properties of these operators are verified with the help of the universal Korovkin‐type property and the order of approximation is calculated by using classical modulus of continuity. Furthermore, weighted ‐statistical convergence and statistical weighted ‐ summability properties of the operators are obtained. Voronovskaja type results are established. The convergence of these operators are also discussed in weighted spaces of functions on the positive semi‐axis and estimate the approximation with the help of weighted modulus of continuity. Moreover, some numerical and graphical examples are also given to justify the results. Lastly, error bounds are calculated associated with the different choice of sequence .

A Generalization of Szasz Operators by Using the Appell Polynomials of Class A(2)

In this paper, as a generalization of Szasz operators, a brand-new sequence of operators including the Appell polynomials of class A2 is introduced. First, the convergence of this new sequence of operators is obtained, and then, some approximation results are presented by using the tools of approximation theory. In addition, an explicit example for this kind of sequence of operators containing Gould–Hopper polynomials is introduced. The error of the approximation of this new sequence of operators to a function is established.

A Note on the Laguerre-Type Appell and Hypergeometric Polynomials

The Laguerre derivative and its iterations have been used to define new sets of special functions, showing the possibility of generating a kind of parallel universe for mathematical entities of this kind. In this paper, we introduce the Laguerre-type Appell polynomials, in particular, the Bernoulli and Euler case, and we examine a set of hypergeometric Laguerre–Bernoulli polynomials. We show their main properties and indicate their possible extensions.

Relating random matrix map enumeration to a universal symbol calculus for recurrence operators in terms of Bessel–Appell polynomials

Maps are polygonal cellular networks on Riemann surfaces. This paper analyzes the construction of closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. The method of construction developed here involves a novel asymptotic symbol calculus for difference operators based on the relation between spectral asymptotics for Hermitian random matrices and asymptotics of orthogonal polynomials with exponential weights. These closed form expressions have a universal character in the sense that they are independent of the explicit valence distribution of the cellular networks within a broad class. Nevertheless the valence distributions may be recovered from the closed form generating functions by a remarkable unwinding identity in terms of Appell polynomials generated by Bessel functions. Our treatment reveals the generating functions to be solutions of nonlinear conservation laws and their prolongations. This characterization enables one to gain insights that go beyond more traditional methods that are purely combinatorial. Universality results are connected to stability results for characteristic singularities of conservation laws that were studied by Caflisch, Ercolani, Hou and Landis, Multi-valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems, Commun. Pure Appl. Math. 46 (1993) 453–499, as well as directly related to universality results for random matrix spectra.

Functional equations related to Appell polynomials and Heun functions

Functional equations related to Appell polynomials and Heun functions

Fock and Hardy spaces: Clifford Appell case

AbstractIn this paper, we study a specific system of Clifford–Appell polynomials and, in particular, their product. Moreover, we introduce a new family of quaternionic reproducing kernel Hilbert spaces in the framework of Fueter regular functions. The construction is based on a general idea which allows us to obtain various function spaces by specifying a suitable sequence of real numbers. We focus on the Fock and Hardy cases in this setting, and we study the action of the Fueter mapping and its range.

Approximation of GBS Type q-Jakimovski-Leviatan-Beta Integral Operators in Bögel Space

In the present article, we introduce the bivariate variant of Beta integral type operators based on Appell polynomials via q-calculus. We study the local and global type approximation properties for these new operators. Next, we introduce the GBS form for these new operators and then study the degree of approximation by means of modulus of smoothness, mixed modulus of smoothness and Lipschitz class of Bögel continuous functions.

Non-separable rotation moment invariants

In this paper, we introduce new rotation moment invariants, which are composed of non-separable Appell moments. We prove that Appell polynomials behave under rotation as monomials, which enables easy construction of the invariants. We show by extensive tests that non-separable moments may outperform the separable ones in terms of recognition power and robustness thanks to a better distribution of their zero curves over the image space.

Densely generated 2D q-Appell polynomials of Bessel type and q-addition formulas

The article aims to introduce a densely generated class of $2D$ $q$-Appell polynomials of Bessel type via generating equation and to establish its $q$-determinant form. It is advantageous to consider the $2D$ $q$-Bernoulli, $2D$ $q$-Roger Szeg\"{o} and $2D$ $q$-Al-Salam Carlitz polynomials of Bessel type as their special members. The $q$-determinant forms and certain $q$-addition formulas are derived for these polynomials. The article concludes with a brief view on discrete $q$-Bessel convolution of the $2D$ $q$-Appell polynomials.

Finding determinant and integral forms of the 2-iterated 2D Appell polynomials

Finding determinant and integral forms of the 2-iterated 2D Appell polynomials

Approximation by Szász-Baskakov operators based on Boas-Buck-type polynomials

This paper concerns with a generalization of Sz?sz-Baskakov operators, which includes Boas-Buck-type polynomials. The convergence properties are studied in weighted space and the rate of convergence is obtained by using weighted modulus of continuity. A Voronovskaya-type theorem is investigated. Also, the theoretical results are demonstrated by choosing the particular cases of Boas-Buck-type polynomials, namely Appell polynomials, Hermite polynomials, Gould-Hopper polynomials, Laguerre polynomials and Charlier polynomials.

Ordinary generating function for a class of Appell polynomials and Stirling symmetric polynomials

Ordinary generating function for a class of Appell polynomials and Stirling symmetric polynomials

Some families of differential equations associated with the 2-iterated 2D Appell and related polynomials

Some families of differential equations associated with the 2-iterated 2D Appell and related polynomials

General Bivariate Appell Polynomials via Matrix Calculus and Related Interpolation Hints

An approach to general bivariate Appell polynomials based on matrix calculus is proposed. Known and new basic results are given, such as recurrence relations, determinant forms, differential equations and other properties. Some applications to linear functional and linear interpolation are sketched. New and known examples of bivariate Appell polynomial sequences are given.

A sequence of Appell polynomials and the associated Jakimovski–Leviatan operators

We are concerned with a special sequence of Appell polynomials, related to the Renyi and Tsallis entropies for the binomial distribution. The generating function is investigated: it is logarithmically convex and has remarkable connections with the modified Bessel function $$I_0(t)$$ and with the index of coincidence for Poisson distribution. The specific form of the Appell polynomials leads to specific properties of the associated Jakimovski–Leviatan operators.

Differential and integral equations for Legendre-Laguerre based hybrid polynomials

UDC 517.9 In this article, a hybrid family of three-variable Legendre – Laguerre – Appell polynomials is explored and their properties including the series expansions, determinant forms, recurrence relations, shift operators, followed by differential, integro-differential and partial differential equations are established. The analogous results for the three-variable Hermite – Laguerre – Appell polynomials are deduced. Certain examples in terms of Legendre – Laguerre – Bernoulli, –E uler and – Genocchi polynomials are constructed to show the applications of main results. A further investigation is performed by deriving homogeneous Volterra integral equations for these polynomials and for their relatives.

HIGHER ORDER APOSTOL-TYPE POLY-GENOCCHI POLYNOMIALS WITH PARAMETERS a, b AND c

In this paper, a new form of poly-Genocchi polynomials is defined by means of poly-logarithm, namely, the Apostol-type poly-Genocchi polynomials of higher order with parameters a, b and c. Several properties of these polynomials are established including some recurrence relations and explicit formulas, which express these higher order Apostol-type poly-Genocchi polynomials in terms of Stirling numbers of the second kind, Apostol-type Bernoulli and Frobenius polynomials of higher order. Moreover, certain differential identity is obtained that leads this new form of poly-Genocchi polynomials to be classified as Appell polynomials and, consequently, draw more properties using some theorems on Appell polynomials. Furthermore, a symmetrized generalization of this new form of poly-Genocchi polynomials is introduced that possesses a double generating function. Finally, the type 2 Apostol-poly-Genocchi polynomials with parameters a, b and c are defined using the concept of polyexponential function and several identities are derived, two of which show the connections of these polynomials with Stirling numbers of the first kind and the type 2 Apostol-type poly-Bernoulli polynomials.

On 2-variable $ q $-Hermite polynomials

<abstract> The quantum calculus has emerged as a connection between mathematics and physics. It has wide applications, particularly in quantum mechanics, analytic number theory, combinatorial analysis, operation theory etc. The $ q $-calculus, which serves as a powerful tool to model quantum computing, has drawn attention of many researchers in the field of special functions and as a result the $ q $-analogues of certain special functions, especially hypergeometric function, 1-variable Hermite polynomials, Appell polynomials etc., are introduced and studied. In this paper, we introduce the 2-variable $ q $-Hermite polynomials by means of generating function. Also, its certain properties like series definition, recurrence relations, $ q $-differential equation and summation formulas are established. The operational definition and some integral representations of these polynomials are obtained. Some examples are also considerd to show the efficacy of the proposed method. Some concluding remarks are given. At the end of this paper, the graphical representations of these polynomials of certain degrees with specified values of $ q $ are given. </abstract>

Обобщённая формула бинома Ньютона и формулы суммирования

В основе работы лежит формула бинома Ньютона и её обобщения на последовательности многочленов биномиального типа. Даны применения к обобщённой проблеме Варинга (Хуа Ло-кен) и проблеме Гильберта – Камке (Г.И.Архипов). Доказана формула Тейлора – Маклорена для многочленов и гладких функций и даны её приложения в численном анализе (решение уравнений методом касательных Ньютона, лемма Гензеля в полныхнеархимедовских полях, приближенное вычисление значений гладких функций в точке). Даётся аналог формулы бинома Ньютона для многочленов Бернулли и доказывается формула Эйлера — Маклорена суммирования значений функции по целым точкам, выведена формула Пуассона суммирования значений функции. Рассмотрены примеры последовательностей многочленов биномиального типа (степени, нижние и верхние факториальныестепени, многочлены Абеля и Лагерра). Найдены биномиальные свойства многочленов Аппеля и Эйлера. Для многочленов и гладких функций от нескольких переменных доказана формула Тейлора, получены многомерные аналоги формул Эйлера – Маклорена и Пуассона суммирования значений функции по решётке. Рассмотрен многомерный аналог этих формул для решётки в многомерном комплексном пространстве. Доказаны ряд свойствпоследовательности многочленов биномиального типа от нескольких переменных.