Some relations satisfied by Hermite-Hermite matrix polynomials

Abstract

The classical Hermite-Hermite matrix polynomials for commutative matrices were first studied by Metwally et al. (2008). Our goal is to derive their basic properties including the orthogonality properties and Rodrigues formula. Furthermore, we define a new polynomial associated with the Hermite-Hermite matrix polynomials and establish the matrix differential equation associated with these polynomials. We give the addition theorems, multiplication theorems and summation formula for the Hermite-Hermite matrix polynomials. Finally, we establish general families and several new results concerning generalized Hermite-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.…”

On generalized two-index Hermite matrix polynomials

“…If A is a matrix in C r r such that Re .´/ > 0 for every eigenvalue´2 .A/; we say a positive stable matrix. Also throughout this paper, we will use a positive stable matrix A in C r r : Then Hermite matrix polynomials are defined by means of the Some other properties of Hermite matrix polynomials can be found in [1,3,7,15,22,29,33].…”