Generalizations of the Bernoulli and Appell polynomials

In this paper, we present a new family of generalized Bernoulli-type polynomials, as well as its numbers. In addition, we obtain some results such as algebraic and differential properties for this new family of Bernoulli-type polynomials. Likewise, the generalized Bernoulli-type polynomials matrix R (α) (x) is introduced. We deduce some product formulae for R (α) (x) and also, the inverse of the Bernoullitype matrix R is determined. Furthermore, we establish some explicit expressions for the Bernoullitype polynomial matrix R(x), which involve the generalized Pascal matrix and finally we study the summation formula of Euler-Maclaurin type and the Riemann zeta function applied to these Bernoullitype polynomials.

Fractal and Fractional

This paper introduces a new type of polynomials generated through the convolution of generalized multivariable Hermite polynomials and Appell polynomials. The paper explores several properties of these polynomials, including recurrence relations, explicit formulas using shift operators, and differential equations. Further, integrodifferential and partial differential equations for these polynomials are also derived. Additionally, the study showcases the practical applications of these findings by applying them to well-known polynomials, such as generalized multivariable Hermite-based Bernoulli and Euler polynomials. Thus, this research contributes to advancing the understanding and utilization of these hybrid polynomials in various mathematical contexts.

Boletín de la Sociedad Matemática Mexicana, 2019

In this article, recurrence relation, differential equations and integral equation for the Laguerre-Gould-Hopper-based Appell polynomials are derived using the factorization method. An analogous study of these results for the Laguerre-Gould-Hopper-based Bernoulli, Euler and Genocchi polynomials is also presented.

European Journal of Pure and Applied Mathematics

Most unifications of the classical or generalized Bernoulli, Euler, and Genocchi polynomials involve unifying any two or all of the three special types of polynomials (see, [1, 4, 9, 18, 19,21, 24–26, 30, 31]). In this paper, we introduce a new class of multiparameter Fubini-type gener-alized polynomials that unifies four families of higher order generalized Apostol-type polynomials such as the Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi, and Apostol-Fubini polynomials. Moreover, we obtain an explicit formula of these unified generalized polynomials in terms of the Gaussian hypergeometric function, and establish several symmetry identities.

Fasciculi Mathematici, 2017

In this paper, we introduce a new class of Hermite multiple-poly-Bernoulli numbers and polynomials of the second kind and investigate some properties for these polynomials. We derive some implicit summation formulae and general symmetry identities by using different analytical means and applying generating functions. The results derived here are a generalization of some known summation formulae earlier studied by Pathan and Khan.

Mathematics

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.

Applied Mathematics and Computation, 2013

The Appell polynomial sequences [5,112] have been studied because of th(ii remarkable applications in theoretical physics, chemistry [94] and several brand ie> of mathematics [7,123] such as the study of polynomial expansions of analytic fiuici'ias, number theory and numerical analysis. The approach to Appell poljaiomial seqimi-es via operational techniques is an easily comprehensible mathematical tool, tliat is important because of the reason that many polynomials arise in physics, chemistr\ md engineering. In last years attention has centered on obtaining a new representation of Appell polynomials. For example, Lehmer [93] illustrated six difTerent approaclus tor representing the sequence of BernouUi polynomials, which is a special case of A)ji ;11 polynomial sequences. The class of Appell sequences defined by generating function (1.4.7) contains a large number of classical polynomial sequences such as the Bernoulli, Euler, Her nue and Laguerre polynomials etc. The Bernoulli polynomials play an important role ;n various expansions and approximation formulae, w^hich are useful both in analytic theoi y of numbers and in classical and numerical analysis. The Bernoulli polynomials S,,(') were discovered by Jacob Bernoulli to evaluate the sum l'' + 2^^ + 3'' + ... + {n-1)^ as kl /Q" Bk{x)dx. These polynomials appear in various problems in the fields of engineeriuf and physics providing their solutions. The Bernoulli polynomials B"(x) are defined by the following generating funct ion The contents of this chapter are published in Applied Mathematics and Computation 219| 17 (2013) 9469-9483.

Advances in Difference Equations, 2013

In this paper, we introduce the extended 2D Bernoulli polynomials by t α (e t-1) α c xt+yt j = ∞ n=0 B (α,j) n (x, y, c) t n n! and the extended 2D Euler polynomials by 2 α (e t + 1) α c xt+yt j = ∞ n=0 E (α,j) n (x, y, c) t n n! , where c > 1. By using the concepts of the monomiality principle and factorization method, we obtain the differential, integro-differential and partial differential equations for these polynomials. Note that the above mentioned differential equations for the extended 2D Bernoulli polynomials reduce to the results obtained in (Bretti and Ricci in Taiwanese J. Math. 8(3): 415-428, 2004), in the special case c = e, α = 1. On the other hand, all the results for the second family are believed to be new, even in the case c = e, α = 1. Finally, we give some open problems related with the extensions of the above mentioned polynomials.

2013

In this paper we use probabilistic methods to derive some results on the generalized Bernoulli and generalized Euler polynomials. Our approach is based on the properties of Appell polynomials associated with uniformly distributed and Bernoulli distributed random variables and their sums.

Kragujevac Journal of Mathematics, 2020

The special polynomials of more than one variable provide new means of analysis for the solutions of a wide class of partial differential equations often encountered in physical problems. Motivated by their importance and potential for applications in a variety of research fields, recently, numerous polynomials and their extensions have been introduced and investigated. In this paper, we introduce a new family of Laguerre-based generalized Hermite-Euler polynomials, which are related to the Hermite, Laguerre and Euler polynomials and numbers. The results presented in this paper are based upon the theory of the generating functions. We derive summation formulas and related bilateral series associated with the newly introduced generating function. We also point out that the results presented here, being very general, can be specialized to give many known and new identities and formulas involving relatively simple numbers and polynomials.