The main aim of this paper is to define and study of a new polynomial, say, two-index Hermite-Hermite matrix polynomials. An explicit representation, a matrix recurrence relation and matrix differential equations of these polynomials are presented. A new expansion of the matrix functions for a wide class of matrices in terms of two-index Hermite-Hermite matrix polynomials is proposed.

In this paper, Tricomi and Hermite-Tricomi matrix functions are introduced starting from the Hermite matrix polynomials. The convergence, radius of regularity, integral form, generating matrix functions, matrix recurrence relations satisfied by these Tricomi matrix functions are derived. Finally, the generating matrix functions, matrix recurrence relations, addition theorems for the Hermite-Tricomi matrix functions are given and matrix differential equations satisfied by them are presented.

The aim of this paper is to define and study of the Gegenbauer matrix polynomials of two variables. An explicit representation, a three-term matrix recurrence relation, differential recurrence relations and hypergeometric matrix representation for the Gegenbauer matrix polynomials of two variables are given. The Gegenbauer matrix polynomials are solutions of the matrix differential equations and expansion of the Gegenbauer matrix polynomials as series of Hermite and Laguerre matrix polynomials of two variables are established.

This paper is motivated by an open problem of Luke’s theorem. We consider the problem of developing a unified point of view on the theory of inequalities of Humbert functions and of their general ratios are obtained. Some particular cases and refinements are given. Finally, we obtain some important results involving inequalities of Bessel and Whittaker’s functions as applications.

The main aim of this paper is to define and study of a new matrix polynomials, say, the Rice’s matrix polynomials. The convergence, radius of regularity, integral form, generating matrix functions and matrix recurrence relations satisfied by these Rice’s matrix polynomials are derived. Furthermore, we study the operation of differential operators of Rice’s matrix polynomials and their applications are presented. The matrix differential equation are obtained by them is presented. Finally, the study of the composition of Rice’s matrix polynomials is investigated.

The main aim of this paper is to define and study of the Chebyshev matrix polynomials. An explicit representation, a three-term matrix recurrence relations for the Chebyshev matrix polynomials are given. These polynomials appear as finite series solutions of second order matrix differential equations. The expansion Chebyshev matrix polynomials in series of Hermite matrix polynomials and the Christoffel formula of summation are established.

The main aim of this paper is to define and study of the Chebyshev matrix polynomials. An explicit representation, a three-term matrix recurrence relations for the Chebyshev matrix polynomials are given. These polynomials appear as finite series solutions of second order matrix differential equations. The expansion Chebyshev matrix polynomials in series of Hermite matrix polynomials and the Christoffel formula of summation are established.

The principal object of this work is to define and study of a new kummer’s matrix function, namely, composite l(m, n)-Kummer’s matrix functionof two complex variables. The radius of regularity and matrix recurrence relationson this function are obtained. The effect of differential operator (D) on thisfunction is investigated and a solution of a certain partial differential equation is established.

The principal object of this work is to define and study of a new kummer’s matrix function, namely, composite l(m, n)-Kummer’s matrix functionof two complex variables. The radius of regularity and matrix recurrence relationson this function are obtained. The effect of differential operator (D) on thisfunction is investigated and a solution of a certain partial differential equation is established.

The principal object of this work is to define and study of a new kummer’s matrix function, namely, composite l(m, n)-Kummer’s matrix functionof two complex variables. The radius of regularity and matrix recurrence relationson this function are obtained. The effect of differential operator (D) on thisfunction is investigated and a solution of a certain partial differential equation is established.