A New Extension of Generalized Hermite Matrix Polynomials

Abstract

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 expansion of these new generalized Hermite matrix polynomials. We also give new series expansions of the matrix functions exp(x B), sin(x B), cos(x B), cosh(x B) and sinh(x B) 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.

On new extensions of the generalized Hermite matrix polynomials

“…Various possible extensions to the matrix framework of the classical families of Laguerre, Hermite, Legendre, Gegenbauer, and Chebyshev polynomials have been widely investigated in the literature (see, for example, [2,3,4,7,10,11,14,16,22,24,25,26,27]). Earlier, the Hermite matrix polynomials and its extensions and generalizations were introduced in [15,21,23,29] for matrices in C N×N , whose eigenvalues are all situated in the right open half-plane.…”

Some relations satisfied by Hermite-Hermite matrix polynomials

“…Important connections between orthogonal matrix polynomials and matrix differential equations of the second order appear in [2], [8], [6], [7]. Extensions to the matrix framework of the classical families of Legendre, Laguerre, Hermite, Chebychev, Hermite-Hermite and Gegenbauer polynomials have been introduced in [1], [9], [10], [12], [14]- [31]. The interest in the family of Hermite polynomials is based on their intrinsic mathematical properties due to which these polynomials have found wideranging applications in physics.…”