Theory of generalized hermite polynomials

International Journal of Mathematics and Mathematical Sciences, 2003

We introduce new families of Hermite polynomials and of Bessel functions from a point of view involving the use of nonexponential generating functions. We study their relevant recurrence relations and show that they satisfy differential-difference equations which are isospectral to those of the ordinary case. We also indicate the usefulness of some of these new families.

AIP Conference Proceedings, 2010

In recent years classical polynomials of a real or complex variable and their generalizations to the case of several real or complex variables have been in a focus of increasing attention leading to new and interesting problems. In this paper we construct higher dimensional analogues to generalized Laguerre and Hermite polynomials as well as some based functions in the framework of Clifford analysis. Our process of construction makes use of the Appell sequence of monogenic polynomials constructed by Falcão/Malonek and stresses the usefulness of the concept of the hypercomplex derivative in connection with the adaptation of the operational approach, developed by Gould et al. in the 60's of the last century and by Dattoli et al. in recent years for the case of the Laguerre polynomials. The constructed polynomials are used to define related functions whose properties show the application of Special Functions in Clifford analysis.

International Journal of Pure and Apllied Mathematics, 2017

The Hermite polynomials represent a powerful tool to investigate the properties of many families of Special Functions. We present some relevant results where the generalized Hermite polynomials of Kampé de Fériet type, simplify the definitions and the operational properties of the two-variable, generalized Bessel functions and their modified. We also discuss a special class of polynomials, recognized as Hermite polynomials, which present a flexible form to describe the two-index, one-variable Bessel functions. By using the generating function method, we will obtain some relations involving these classes of Hermite polynomials and we can also compare them with the Humbert polynomials and functions.

Fractal and Fractional

In this article, the recurrence relations and shift operators for multivariate Hermite polynomials are derived using the factorization approach. Families of differential equations, including differential, integro–differential, and partial differential equations, are obtained using these operators. The Volterra integral for these polynomials is also discovered.

We discuss the theory of multivariable multiindex Bessel functions (B.F.) and Hermite polynomials (H.P.) using the generating function method. We derive addition and multiplication theorems and discuss how generalized H.P. can be exploited as a useful complement to the theory of B.F.. We also discuss the importance of the Poisson-Charlier polynomials in the context of multiindex special functions.

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.

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.

Bulletin of the Malaysian Mathematical Sciences Society, 2014

The Hermite matrix polynomials have been generalized in a number of ways and many of these generalizations have been shown to be important tools in applications. In this paper we introduce a new generalization of the Hermite matrix polynomials and present the recurrence relations and the expansions of these new generalized Hermite matrix polynomials. We also give new series expansions of the matrix functions exp(xB), sin(xB), cos(xB), cosh(xB) and sinh(xB) in terms of these generalized Hermite matrix polynomials and thus prove that many of the seemingly different generalizations of the Hermite matrix polynomials may be viewed as particular cases of the two-variable polynomials introduced here. The generalized Chebyshev and Legendre matrix polynomials have also been introduced in this paper in terms of these generalized Hermite matrix polynomials.

International Journal of Contemporary Mathematical Sciences, 2014

The intended objective of this paper is to extend the Hermite polynomials based on hypergeometric functions and to prove basic properties of the extended Hermite polynomials.

Journal of Functional Analysis 266 (5), pp. 2910-2920., 2014

We consider a new generalization of Hermite polynomials to the case of several variables. Our construction is based on an analysis of the generalized eigenvalue problem for the operator ∂ Ax + D, acting on a linear space of polynomials of N variables, where A is an endomorphism of the Euclidean space R N and D is a second order differential operator. Our main results describe a basis for the space of Hermite-Jordan polynomials.