Generalized Form of Hermite Matrix Polynomials via the Hypergeometric Matrix Function ()
1. 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, 
, have been introduced and studied in [2] [3] where 
involves a parameter 
whose eigenvalues are all situated in the open right-hand half of the complex plane. The two-variable Hermite matrix polynomials, 
, have been presented in [4] as an extension of
. Moreover, some properties and other generalizations of 
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, 
denotes the set of complex matrices of size 
and the matrices 
and 
in 
denote the matrix identity and the zero matrix of order
, respectively. For a matrix 
in
, its spectrum 
denotes the set of all eigenvalues of
. We say that a matrix 
in 
is a positive stable if
(1)
If 
and 
are holomorphic functions of the complex variable
, which are defined in an open set 
of the complex plane and 
is a matrix in 
with
, then from the properties of the matrix functional calculus ([14] , p. 558), it follows that
.
If 
is the complex plane cut along the negative real axis and Log(z) denotes the principle logarithm of zthen 
represents
. If A is a matrix in 
with
, then 
denotes the image by 
of the matrix functional calculus acting on the matrix
.
Let 
be a matrix in 
which satisfies the condition (1). The two-variable Hermite matrix polynomials [2VHMPs] are generated by [4]
(2)
and are defined by the series
(3)
where 
is the standard floor function which maps a real number 
to its next smallest integer.
It is therefore evident, for
, that
(4)
where 
is the Hermite matrix polynomials as given in [2] . Furthermore,
According to [4] , we have
(5)
(6)
Also, the 2VHMPs appear as a solution of the second order matrix differential equation in the form
(7)
and satisfy the three terms recurrence relationship
(8)
with 
and
.
From (5), the relation (8) gives
(9)
Iteration (9) yields a another representation of the 2VHMPs in the form
(10)
Another remarkable representation of the 2VHMPs, which is due essentially to ([4] , Theorem 7), has the elegant form:
(11)
Applying (11) provides the formula
(12)
In fact, the addition and multiplication theorems are
(13)
and
(14)
If 
and 
are matrices in 
for 
and
, then it follows that [10] [15] [16] :
(15)
and
(16)
2. 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
. Then, for any matrix 
in
, the image of 
acting on
, denoted by 
is a well-defined matrix. Furthermore, if
(17)
then 
is invertible, its inverse coincides with 
and it follows that ([17] , p. 253)
(18)
with
.
If A is a positive stable matrix in
, then the gamma matrix function, 
, is well defined as [18]
(19)
From ([19] , p. 206), we have
(20)
Definition 2.1 [15] Let 
and 
be two non-negative integers. The generalized hypergeometric matrix function is defined in the form:
(21)
where 
and 
are matrices in 
such that the matrices 
satisfy the condition (17).
According to [15] , it follows that:
• If
, then the power series (21) converges for all finite
.
• If
, then the power series (21) is absolutely convergent for 
and diverges for
.
• If 
then the power series (21) diverges for
.
With 
and 
in (21), one gets the following relation due to ([3] , p. 213)
(22)
which can be written by (19) and (20) in the form
(23)
For 
and 
in (21), we obtain the hypergeometric matrix function as given in [3] in the form
(24)
Moreover, the confluent hypergeometric matrix function is well defined for all finite
, when
, in the form
(25)
One can easily get the following result.
Proposition 2.2
(26)
and
(27)
In [20] , the following theorem was proved:
Theorem 2.3 Let A and B be two matrices in 
such that 1. 
and 
are positive stable2.
3. 
is invertible for all
.
Then for a positive integer 
the following holds
(28)
Indeed, by (20) we can rewrite the formula (28) in the form
(29)
A matrix version of Kummer’s first formula for the confluent hypergeometric matrix function is presented in the following theorem:
Theorem 2.4 Let A and B be two matrices in 
satisfy the conditions of Theorem 2.3. Then
(30)
Proof. From (15) and (25) we have
(31)
By (29) and taking into account the conditions of Theorem 2.3 we find
(32)
and so (30) follows. □
3. Generalized Chebyshev Matrix Polynomials
In [20] , the Chebyshev matrix polynomials of the first kind 
was defined by
(33)
From (20) and (25) with the use of
(34)
we give an integral representation of 
in the form
(35)
The generalized Chebyshev matrix polynomials of the second kind [GCMPs] are defined by the series [4]
(36)
and specified by the integral representation
(37)
According to (14), the integral representation (37) becomes
(38)
The use of the relations (5) and (8) in (37) yields the differential recurrence relation
According to [4] , we have 
and
, where 
is the Chebyshev matrix polynomials of the second kind [CMPs].
As a direct consequent of ([4] , Lemma 5), we state the following result.
Proposition 3.1 For a real number
, it follows that
(39)
where 
and 
Let us now introduce the two-variable and two-index Chebyshev matrix polynomials of two matrices [2V2ICMP] through the integral representation
(40)
where 
and 
are two matrices in 
satisfy the condition (1). From (3) and (34) we obtain that
(41)
Indeed, by (14), the integral representation (40) becomes
(42)
It is worthy to mention that, on taking 
or
, 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
(43)
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
(44)
4. 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 
and 
be two matrices in 
such that 
satisfies the condition (1) and 
satisfies the condition (17). We can define the GHMPs in the form:
(45)
where
(46)
Note that, by (24), the expression (46) can be written in the form
(47)
When 
is the zero matrix, then the GHMPs reduce to the two-variable Hermite matrix polynomials, 
, 
In view of (19), (20) and (34), the expression (46) can be also written in the following integral representation
(48)
It is clear that
and
By using (19), (20) and (34), the formula (45) leads to
Hence, by (3), we obtain the integral representation of the GHMPs in the form
(49)
In view of (12), the integral representation (49) becomes
which, by (37), provides the following form by means of the generalized Chebyshev matrix polynomials
(50)
Thus, by exploiting (19), (20) and (36) in (50), one gets
By (11) and (6), it follows that
Hence from (25) we arrive at the following representation of the GHMPs
(51)
The use of the second order matrix differential Equation (7) in the integral representation (49) gives
which, with the help of (5), obtaining the differential recurrence relation
