Generalized Form of Hermite Matrix Polynomials via the Hypergeometric Matrix Function

The object of this paper is to present a new generalization of the Hermite matrix polynomials by means of the hypergeometric matrix function. An integral representation, differential recurrence relation and some other properties of these generalized forms are established here. Moreover, some new properties of the Hermite and Chebyshev matrix polynomials are obtained. In particular, the two-variable and two-index Chebyshev matrix polynomials of two matrices are presented.


Introduction
Special functions have been developed deeply in the last decades to special matrix functions due to their applications in certain areas of statistics, physics and engineering.The Laguerre and Hermite matrix polynomials are introduced in [1] as examples of right orthogonal matrix polynomial sequences for appropriate right matrix moment functionals of integral type.The Hermite matrix polynomials, ( ) , , n H x y A , have been presented in [4] as an extension of ( ) , n H x A .Moreover, some properties and other generalizations of ( ) , n H x A are given in [5]- [11].As one of qualitative properties of the two-variable Hermite matrix polynomials, the Chebyshev matrix polynomials of the second kind are introduced in [4], see also [12] [13].
The main aim of this paper is to consider a new generalization of the Hermite matrix polynomials and to derive some properties for the Hermite and Chebyshev matrix polynomials.The structure of this paper is the following.This section summarizes previous results essential in the rest of the paper and gives the development of the two-variable Hermite matrix polynomials.A matrix version of Kummer's first formula for the confluent hypergeometric matrix function is derived in Section 2. In Section 3, the addition theorem and three terms recurrence relation for the Chebyshev matrix polynomials of the second kind are obtained and further we introduce and study the two-variable and two-index Chebyshev matrix polynomials of two matrices.Finally, Section 4 deals with the study of the Generalized Hermite matrix polynomials by means of the hypergeometric matrix function.
In what follows, r r ×  denotes the set of complex matrices of size r r × and the matrices I and θ in r r ×  denote the matrix identity and the zero matrix of order r , respectively.For a matrix A in r r ×  , its spectrum ( ) A σ denotes the set of all eigenvalues of A .We say that a matrix If ( ) f z and ( ) g z are holomorphic functions of the complex variable z , which are defined in an open set Ω of the complex plane and A is a matrix in r r ×  with ( ) ⊂ Ω , then from the properties of the matrix functional calculus ( [14], p. 558), it follows that ( ) ( ) ( ) ( ) D is the complex plane cut along the negative real axis and Log(z) denotes the principle logarithm of z, the image by 1 2  z of the matrix functional calculus acting on the matrix A .Let A be a matrix in r r ×  which satisfies the condition (1).The two-variable Hermite matrix polynomials [2VHMPs] are generated by [4] ( ) and are defined by the series where ν     is the standard floor function which maps a real number ν to its next smallest integer.
It is therefore evident, for 1 y = , that , n H x A is the Hermite matrix polynomials as given in [2].Furthermore, (

H x y A y H x y A =
According to [4], we have Also, the 2VHMPs appear as a solution of the second order matrix differential equation in the form , , 0.
and satisfy the three terms recurrence relationship , ; 2 with , , .
Iteration ( 9) yields a another representation of the 2VHMPs in the form .
Another remarkable representation of the 2VHMPs, which is due essentially to ([4], Theorem 7), has the elegant form: .
In fact, the addition and multiplication theorems are ( ) ( ) and

The Confluent Hypergeometric Matrix Function
In this section, the confluent hypergeometric matrix function is given.For the sake of clarity in the presentation, we recall some concepts and results related to the generalized hypergeometric matrix functions, that may be found in [15] [18] [19].
The reciprocal gamma function denoted by ( ) ( ) is an entire function of the complex variable z .Then, for any matrix A in r r ×  , the image of ( ) then ( ) is invertible, its inverse coincides with ( ) with ( ) 0 If A is a positive stable matrix in r r ×  , then the gamma matrix function, ( ) , is well defined as [18] ( ) From ( [19], p. 206), we have Definition 2.1 [15] Let p and q be two non-negative integers.The generalized hypergeometric matrix function is defined in the form: where i A and j B ( ) such that the matrices j B satisfy the condition (17).
According to [15], it follows that: • If p q ≤ , then the power series (21) converges for all finite z .
• If which can be written by ( 19) and ( 20) in the form ( ) ( ) ( ) For 2 p = and 1 q = in (21), we obtain the hypergeometric matrix function as given in [3] in the form Moreover, the confluent hypergeometric matrix function is well defined for all finite z , when In [20], the following theorem was proved: Theorem 2.

F nI A B B A nI B nI B A B
Indeed, by (20) we can rewrite the formula (28) in the form , ; ;1 .
A matrix version of Kummer's first formula for the confluent hypergeometric matrix function is presented in the following theorem: Proof.From ( 15) and (25) we have By ( 29) and taking into account the conditions of Theorem 2.3 we find and so (30) follows.□

Generalized Chebyshev Matrix Polynomials
In [20], the Chebyshev matrix polynomials of the first kind ( ) From ( 20) and ( 25) with the use of 0 !e d , we give an integral representation of ( ) The generalized Chebyshev matrix polynomials of the second kind [GCMPs] are defined by the series [4] ( and specified by the integral representation According to (14), the integral representation (37) becomes The use of the relations ( 5) and ( 8) in (37) yields the differential recurrence relation .
According to [4], we have As a direct consequent of ([4], Lemma 5), we state the following result.Proposition 3.1 For a real number where A and B are two matrices in r r ×  satisfy the condition (1).From ( 3) and (34) we obtain that Indeed, by ( 14), the integral representation (40) becomes ( ) It is worthy to mention that, on taking 0 n = or 0 m = , the Equations ( 40), ( 41) and (42) of the 2V2ICMP reduce to the Equations ( 38), ( 36) and (37) of the [CMPs], respectively.
It is evident that the formula (37) provides ( ) Thus, by applying ( 13) in (43), we obtain which, in view of ( 14), one gets This, by the formula (42), leads to the addition theorem for the Chebyshev matrix polynomials of the second kind in the form

Generalized Hermite Matrix Polynomials
By using the hypergeometric matrix function it is convenient to consider a new generalized form of the Hermite matrix polynomials.The generalized Hermite matrix polynomials [GHMPs] of two matrices and two variables are presented here.Let A and B be two matrices in r r ×  such that A satisfies the condition (1) and B satisfies the condition (17).We can define the GHMPs in the form: where ( ) , , ; ; .
Note that, by (24), the expression (46) can be written in the form when B is the zero matrix, then the GHMPs reduce to the two-variable Hermite matrix polynomials, ..e i , ( , , , , . In view of ( 12), the integral representation (49) becomes which, by (37), provides the following form by means of the generalized Chebyshev matrix polynomials

B n n H x y A B nI y H x y A y
The use of the second order matrix differential Equation (7) all situated in the open right-hand half of the complex plane.The two-variable Hermite matrix polynomials, ( ) 0 k ≥ , then it follows that[10] [15] [16]: then the power series (21) is absolutely convergent for 1 z < and diverges for 1 z > .• If p q ≤ then the power series (21) diverges for 0 z ≠ .With 1 p = and 0 q = in (21), one gets the following relation due to ([3], p. 213)

Theorem 2 . 4
Let A and B be two matrices in r r ×  satisfy the conditions of Theorem 2.3.Then A is the Chebyshev matrix polynomials of the second kind [CMPs].
Let us now introduce the two-variable and two-index Chebyshev matrix polynomials of two matrices [2V2ICMP] through the integral representation (

3
Let A and B be two matrices in r r in the integral representation (49) gives