The main aim of this paper is to define and study a new confluent hypergeometric matrix function, say, (m,n)-Kummer's matrix function of two complex variables. The radius of regularity and recurrence relation on this function is established. The effect of differential operator on this function is investigated. We study the operation of a differential operator on the Hadamard product of two (m,n)-Kummer's matrix functions. Finally, we define the composite (m,n)-Kummer's matrix functions and the effect of differential operator on this function is investigated

The main aim of this paper is to define a new polynomial, say, pseudo hyperbolic matrix functions, pseudo Hermite matrix polynomials and to study their properties. Some formulas related to an explicit representation, matrix recurrence relations are deduced, differential equations satisfied by them is presented, and the important role played in such a context by pseudo Hermite matrix polynomials are underlined.

In this paper, we introduce a new generalization of the Hermite matrix polynomials expansions of some relevant matrix functions appearing in the solution of differential systems. An explicit representation and an expansion of the matrix exponential in a series of these matrix polynomials are obtained. Properties of Hermite matrix polynomials such as the recurrence formula permit an efficient computations of matrix functions are established. A new expansion of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.

In this paper the definition of Hermite-Hermite matrix polynomials is introduced starting from the Hermite matrix polynomials. An explicit representation, a matrix recurrence relation for the Hermite-Hermite matrix polynomials is given and differential equations satisfied by them are presented. A new expansion of the matrix exponential for a wide class of matrices in terms of Hermite-Hermite matrix polynomials are proposed.