Some properties of appell polynomials (original) (raw)
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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.
Filomat, 2014
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.
A set of finite order differential equations for the Appell polynomials
Journal of Computational and Applied Mathematics, 2014
Let {R n (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 L (x) n,k ∞ n=0 such that L (x) n,k (R n (x)) = λ n,k R n (x), where λ n,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.
Differential and integral equations for the 2-iterated Appell polynomials
Journal of Computational and Applied Mathematics, 2016
The main goal of this article is to derive differential and associated integral equations for some 2-iterated and mixed type special polynomial families. By using the factorization method, the recurrence relations and differential equations are derived for the 2-iterated Bernoulli, 2-iterated Euler and Bernoulli-Euler polynomials. The Volterra integral equations for these polynomial families are also established. The graphical representation of these 2-iterated and mixed polynomials is presented. The zeros of these polynomials are investigated for certain values of the index n using numerical computation. The approximate solutions of the real zeros of these polynomials are also given.
Hermite-based Appell polynomials: Properties and applications
Journal of Mathematical Analysis and Applications, 2009
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.
Generalizations of the Bernoulli and Appell polynomials
Abstract and Applied Analysis, 2004
We first introduce a generalization of the Bernoulli polynomials, and consequently of the Bernoulli numbers, starting from suitable generating functions related to a class of Mittag-Leffler functions. Furthermore, multidimensional extensions of the Bernoulli and Appell polynomials are derived generalizing the relevant generating functions, and using the Hermite-Kampé de Fériet (or Gould-Hopper) polynomials. The main properties of these polynomial sets are shown. In particular, the differential equations can be constructed by means of the factorization method.
2-Iterated Appell polynomials and related numbers
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.
Some properties of Hermite based Appell matrix polynomials
Tbilisi Mathematical Journal, 2017
In this paper, the Hermite based Appell matrix polynomials are introduced by using certain operational methods. Some properties of these polynomials are considered. Further, some results involving the 2D Appell polynomials are established, which are proved to be useful for the derivation of results involving the Hermite based Appell matrix polynomials.
Generalized Hermite-based Apostol-Euler Polynomials and Their Properties
2020
The aim of this paper is to study certain properties of generalized Apostol-Hermite-Euler polynomials with three parameters. We have shown that there is an intimate connection between these polynomials and established their elementary properties. We also established some identities by applying the generating functions and deduce their special cases and applications