Series with Hermite polynomials and applications (original) (raw)
A New Class of Generalized Polynomials Associated with Hermite and Euler Polynomials
Mediterranean Journal of Mathematics, 2015
Motivated by their importance and potential for applications in certain problems in number theory, combinatorics, classical and numerical analysis, and other fields of applied mathematics, a variety of polynomials and numbers with their variants and extensions have recently been introduced and investigated. In this paper, we aim to introduce generalized Laguerre-Bernoulli polynomials and investigate some of their properties such as explicit summation formulas, addition formulas, implicit formulas, and symmetry identities. Relevant connections of the results presented here with those relatively simple numbers and polynomials are considered.
arXiv: Classical Analysis and ODEs, 2019
The aim of this paper is to construct generating functions for new families of special polynomials including the Appel polynomials, the Hermite-Kampe de Feriet polynomials, the Milne-Thomson type polynomials, parametric kinds of Apostol type numbers and polynomials. Using Euler's formula, relations among special functions, Hermite-type polynomials, the Chebyshev polynomials and the Dickson polynomials are given. Using generating functions and their functional equations, various formulas and identities are given. With help of computational formula for new families of special polynomials, some of their numerical values are given. Using hypegeometric series, trigonometric functions and the Euler's formula, some applications related to Hermite-type polynomials are presented. Finally, further remarks, observations and comments about generating functions for new families of special polynomials are given.
Multiple-Poly-Bernoulli Polynomials of the Second Kind Associated with Hermite Polynomials
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.
arXiv: Mathematical Physics, 2018
For a sequence P=(pn(x))n=0inftyP=(p_n(x))_{n=0}^{\infty}P=(pn(x))n=0infty of polynomials pn(x)p_n(x)pn(x), we study the KKK-tuple and LLL-shifted exponential lacunary generating functions mathcalGK,L(lambda;x):=sumn=0inftyfraclambdann!pncdotK+L(x)\mathcal{G}_{K,L}(\lambda;x):=\sum_{n=0}^{\infty}\frac{\lambda^n}{n!} p_{n\cdot K+L}(x)mathcalGK,L(lambda;x):=sumn=0inftyfraclambdann!pncdotK+L(x), for K=1,2dotscK=1,2\dotscK=1,2dotsc and L=0,1,2dotscL=0,1,2\dotscL=0,1,2dotsc. We establish an algorithm for efficiently computing mathcalGK,L(lambda;x)\mathcal{G}_{K,L}(\lambda;x)mathcalGK,L(lambda;x) for generic polynomial sequences PPP. This procedure is exemplified by application to the study of Hermite polynomials, whereby we obtain closed-form expressions for mathcalGK,L(lambda;x)\mathcal{G}_{K,L}(\lambda;x)mathcalGK,L(lambda;x) for arbitrary KKK and LLL, in the form of infinite series involving generalized hypergeometric functions. The basis of our method is provided by certain resummation techniques, supplemented by operational formulae. Our approach also reproduces all the results previously known in the literature.
A new type of Hermite matrix polynomial series
Quaestiones Mathematicae, 2017
Conventional Hermite polynomials emerge in a great diversity of applications in mathematical physics, engineering, and related fields. However, in physical systems with higher degrees of freedom it will be of practical interest to extend the scalar Hermite functions to their matrix analogue. This work introduces various new generating functions for Hermite matrix polynomials and examines existence and convergence of their associated series expansion by using Mehler's formula for the general matrix case. Moreover, we derive interesting new relations for even-and odd-power summation in the generatingfunction expansion containing Hermite matrix polynomials. Some new results for the scalar case are also presented.
Multifarious Implicit Summation Formulae of Hermite-Based Poly-Daehee Polynomials
In this paper, we introduce the generating function of Hermite-based poly-Daehee numbers and polynomials. By making use of this generating function, we investigate some new and interesting identities for the Hermite-based poly-Daehee numbers and polynomials including recurrence relations, addition property and correlations with poly-Bernoulli polynomials of second kind. We then derive diverse implicit summation formula for Hermite-based poly-Daehee numbers and polynomials by applying the series manipulation methods.
A Note on a Special Class of Hermite Polynomials
International Journal of Pure and Apllied Mathematics, 2015
This paper is devoted to the description of a special class of Hermite polynomials of five variables. It can be seen as an extension of the generalized vectorial Hermite polynomials of type H m,n (x, y) and at the same time as a generalization of the Gould-Hopper Hermite polynomials of type H n (x, y). We use the five-variable Hermite polynomials to derive reformulations of the well known operational relations satisfied from the generalized Hermite polynomials of different types.
A new generalization of Apostol type Hermite-Genocchi polynomials and its applications
SpringerPlus, 2016
By using the modified Milne-Thomson's polynomial given in Araci et al. (Appl Math Inf Sci 8(6):2803-2808, 2014), we introduce a new concept of the Apostol Hermite-Genocchi polynomials. We also perform a further investigation for aforementioned polynomial and derive some implicit summation formulae and general symmetric identities arising from different analytical means and generating functions method. The results obtained here are an extension of Hermite-Bernoulli polynomials (Pathan and Khan in Mediterr J Math 12:679-695, 2015a) and Hermite-Euler polynomials (Pathan and Khan in Mediterr J Math 2015b, doi:10.1007/s00009-015-0551-1) to Apostol type Hermite-Genocchi polynomials defined in this paper.