Theory of generalized hermite polynomials (original) (raw)
Some Families of Differential Equations Associated with Multivariate Hermite Polynomials
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.
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.
ON A GENERALIZATION OF 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.
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 New Extension of Generalized Hermite Matrix Polynomials
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.
Monogenic generalized hermite polynomials and associated hermite-bessel functions
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.
A New Class of Hermite-Konhauser Polynomials together with Differential Equations
Kyungpook mathematical journal, 2010
It is shown that an appropriate combination of methods, relevant to operational calculus and to special functions, can be a very useful tool to establish and treat a new class of Hermite and Konhauser polynomials. We explore the formal properties of the operational identities to derive a number of properties of the new class of Hermite and Konhauser polynomials and discuss the links with various known polynomials.
Numerical construction of the generalized Hermite polynomials
Publikacije Elektrotehnickog fakulteta - serija: matematika, 2003
In this paper we are concerned with polynomials orthogonal with respect to the generalized Hermite weight function w(x) = |x ? z|? exp(?x2) on R, where z?R and ? > ? 1. We give a numerically stable method for finding recursion coefficients in the three term recurrence relation for such orthogonal polynomials, using some nonlinear recurrence relations, asymptotic expansions, as well as the discretized Stieltjes-Gautschi procedure.
The Hermite polynomials and the Bessel functions from a general point of view
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.