Some applications of the Hermite matrix polynomials series expansions

On Chebyshev matrix polynomials, matrix differential equations and their properties

“…Hermite and Laguerre matrix polynomials were introduced and studied in [2,3,5,15,16,18,21,[25][26][27][28][29][30][31]. Important connections between orthogonal matrix polynomials and matrix differential equations of the second order appear in [4,10,11,22].…”

“…Some differential recurrence relations, in particular Chebyshev's matrix differential equation is established in Sect. 3. Expansion of the series and a connection between Chebyshev's and Hermite's matrix polynomials recently introduced in Section 4.…”

“…If f (z) and g(z) are holomorphic functions of the complex variable z, which are defined in an open set of the complex plane, and if A is a matrix in C N ×N such that σ (A) ⊂ , then from the properties of the matrix functional calculus [3,15], it follows that…”

“…It has been seen by Defez and Jódar [3] that, for matrices A(k, n) and B(k, n) are matrices in C N ×N for n ≥ 0, k ≥ 0, the following relations are satisfied…”

“…For the sake of clarity, we recall that if A is a matrix in C N ×N satisfies the condition (1.2), than the expansion of x n I in a series of Hermite matrix polynomials has been given in [3,10] in the from…”