Analytical Properties of the Generalized Heat Matrix Polynomials Associated with Fractional Calculus

In the past few decades, matrix versions of the orthogonal polynomials have attracted a lot of research interest due to their close relations and various applications in many areas of mathematics, statistics, physics, and engendering, for example, see [1–11]. The recent advances of fractional order calculus (FOC) are dominated by its multidisciplinary applications. Moreover, special functions of fractional order calculus have many applications in various areas of mathematical analysis, probability theory, control systems, and engineering (see, for example, [12–15]). Moreover, the development of fractional calculus associated with special matrix functions and polynomials has been investigated by many researchers, for example, the recent works [16–22]. Among these classical polynomials are the heat polynomials (also designated as Temperature polynomials) that are polynomial solutions of the heat equation and also are particularly useful in solving the Cauchy problem (see [23– 26]). Special functions, such as the confluent hypergeometric function, integral error functions, and Laguerre polynomials, have a close link with the generalized heat polynomials introduced [27–29]. Further, the generalized heat polynomials are mainly used to construct an approximate solution of a given problem as a linear combination of the polynomials. Such solution satisfies the governing equation and other equations (cf., e.g., [30–35]). In our investigation here, we define a generalized heat matrix polynomial HPmðT ; ξ, vÞ. We then establish certain generating matrix functions, finite sum formulas, Laplace transforms, and fractional calculus operators for these polynomials in Sections 3, 4, 5, and 6, respectively. Further, some interesting special cases and concluding remarks of our main results are pointed out in Section 7.

The recent advances of fractional order calculus (FOC) are dominated by its multidisciplinary applications. Moreover, special functions of fractional order calculus have many applications in various areas of mathematical analysis, probability theory, control systems, and engineering (see, for example, [12][13][14][15]).
Among these classical polynomials are the heat polynomials (also designated as Temperature polynomials) that are polynomial solutions of the heat equation and also are particularly useful in solving the Cauchy problem (see [23][24][25][26]). Special functions, such as the confluent hypergeometric function, integral error functions, and Laguerre polynomials, have a close link with the generalized heat polynomials intro-duced [27][28][29]. Further, the generalized heat polynomials are mainly used to construct an approximate solution of a given problem as a linear combination of the polynomials. Such solution satisfies the governing equation and other equations (cf., e.g., [30][31][32][33][34][35]).
In our investigation here, we define a generalized heat matrix polynomial ℍℙ m ðT ; ξ, vÞ. We then establish certain generating matrix functions, finite sum formulas, Laplace transforms, and fractional calculus operators for these polynomials in Sections 3, 4, 5, and 6, respectively. Further, some interesting special cases and concluding remarks of our main results are pointed out in Section 7.

Preliminaries
In this section, we give some basic definitions and terminologies; for more details, we can be referred to [36,37].
Here and through the work, let ℂ d×d be the vector space of all the square matrices of order d ∈ ℕ, (ℕ is the set of all positive integers) whose entries are in the set of complex numbers ℂ. For a E ∈ ℂ d×d , let σðEÞ be the set of all eigenvalues of E which is called the spectrum of E. We have which implyμðEÞ = −μð−EÞ. Here, μðEÞ is called the spectral abscissa of E, and the matrix E is said to be positive stable if μðEÞ > 0. Further, let I and 0 denote the identity and zero matrices corresponding to a square matrix of any order, respectively. If E is a positive stable matrix in ℂ d×d , then the gamma matrix function ΓðEÞ is well defined as follows (cf., e.g., [11,38,39]): Moreover, if E is a matrix in ℂ d×d which gratifies then ΓðEÞ is invertible, its inverse coincides with Γ −1 ðEÞ. Under condition (3), we can write the following Pochhammer matrix symbol Let r, k ∈ ℕ 0 . Also let ðSÞ r and ðQÞ k be arrays of r commutative matrices S 1 , S 2 , ⋯, S r and k commutative matrices Q 1 , Q 2 , ⋯, Q k in ℂ d×d , respectively, such that Q k + nI are invertible for 1 ≤ d ≤ k and all n ∈ ℕ 0 . Then, the generalized hypergeometric matrix function r F k ððSÞ r ; ðQÞ k ; zÞðz ∈ ℂÞ can be defined by (see, e.g., [11,39]) In particular, the hypergeometric matrix function 2 F 1 ðS, P ; C ; zÞ ≡ FðS, P ; C ; zÞ is defined by for matrices R, P, C in ℂ d×d such that C + nI is invertible for all n ∈ ℕ 0 . Also, note that for r = 1, k = 0 in (9), we have the Binomial type matrix function 1 F 0 ðR;−;zÞ as follows: Let E be a positive stable invertible matrix in ℂ d×d : Then, the n th Laguerre matrix polynomial is defined in the form (see, e.g., [11,40]) The Laplace transform of f ðξÞ is defined by [7].
provided that the improper integral exists.

Lemma 1.
(see [7]). Let S be a positive stable invertible matrix in ℂ d×d . Then, we have

Generalized Heat Matrix Polynomial and Generating Matrix Functions
A generalized heat matrix polynomial is defined in (11) below; then, a family of generating matrix functions are proposed, see Theorem 4 and Theorem 8 of this section.
Definition 2. Let T be a positive stable matrix in the complex space C d×d satisfying the spectral condition (3). Then, we define a generalized heat matrix polynomial of degree m ∈ ℕ 0 in the following explicit form: where L T m ðξÞ is the Laguerre matrix polynomial in (8).

Remark 3. Note that
and that for the scalar case d = 1, taking T = a and a > 0, the m th polynomial ℍℙ m ða ; ξ, vÞ coincides with the classical scalar generalized heat polynomial, see [24,26,33]. Further, the ordinary heat polynomial defined in [23], when T = 0; ℍℙ m ð0 ; ξ, vÞ = υ 2m ðξ, vÞ: Proof. For convenience, suppose that the left-hand side of (13) is denoted by J. According to the series expression of (11) and (7) to J, we find that Upon using the relation (6), the last equality evidently leads us to the required result.

Corollary 5.
For ℍℙ m ðT ; ξ, vÞ, the following generating matrix function holds true Remark 6. The Bessel matrix function J R ðzÞ, for a positive stable matrix R ∈ C d×d , is expressible in terms of hypergeometric matrix function as follows (see, e.g., [11,41]) Thus, by applying the relation (16) to (15) in Corollary 5, we can deduce the following Corollary.

Corollary 7.
For ℍℙ m ðT ; ξ, vÞ, the following holds true Theorem 8. Let ξ ∈ ℂ, v > 0, m, l ∈ ℕ 0 and T be a positive stable matrix in C d×d such that T + nI is invertible for all n ∈ ℕ 0 : The following relation holds true Proof. Follows by induction or by the successive application of Theorem 4 when R = ðT + 1/2IÞ: The details are omitted.

Corollary 9.
For l = 0 in Theorem 8, the following holds true Remark 10. The special cases of (18) and (19) when d = 1 are seen to yield the classical generating functions of the generalized Heat polynomials (see [24,33]).

Finite Sums
Here, various finite sums of ℍℙ m ðT ; ξ, vÞ can be obtained in the following results.
Theorem 11. Let ξ, z ∈ ℂ, v > 0, m ∈ ℕ 0 , and T be a positive stable matrix in C d×d such that T + nI is invertible for all n ∈ ℕ 0 : Then, we have Proof. From (15) and the following fact We thus find that Comparing the coefficient of t m on both side, we thus get the required a finite sum formula (20).

Journal of Function Spaces
Theorem 12. Let ξ ∈ ℂ, v > 0, m ∈ ℕ 0 , also let T and R be positive stable matrices in C d×d such that T + nI and R + nI are invertible for all n ∈ ℕ 0 : Then, we have Proof. By using the series (11) and Theorem 4 with applying to Kummer's matrix formula (see [5]), we observe that Equating the coefficient of t m on both sides, we thus arrive at the desired result (23).

Laplace Transforms
Here, Laplace integral transforms of the generalized heat matrix polynomials are derived as follows.
Theorem 13. Let ξ ∈ ℂ, v > 0, RðλÞ > 0, m ∈ ℕ 0 . Also, let T and A be a positive stable matrices in C d×d such that T + nI is invertible for all n ∈ ℕ 0 : The following Laplace transform formula hold Proof. Making a particular use of (9) with (11), (6) and applying to Lemma 1, yields our desired result (25) in Theorem 13. The details are omitted.
A similar procedure yields the following Laplace integral transforms. So we prefer to omit the proofs.

Theorem 14.
Let ξ ∈ ℂ, v > 0, RðλÞ > 0, m ∈ ℕ 0 . Also, let T and A be a positive stable matrices in C d×d such that T + nI is invertible for all n ∈ ℕ 0 : The following Laplace transform formula hold Theorem 15. Let ξ ∈ ℂ, v > 0, RðλÞ > 0, m ∈ ℕ 0 . Also, let T and A be a positive stable matrices in C d×d such that T + nI is invertible for all n ∈ ℕ 0 : The following Laplace transform formula hold The above Theorems lead to the following special cases.

Corollary 16.
For T is a positive stable matrix in C d×d , and RðλÞ > 0, then we have the following Laplace transforms

Fractional Calculus Approach
Here, we consider the Riemann. Liouville fractional integral and derivative operators I γ and D γ z of order γ ∈ C, RðγÞ > 0, respectively (see, for details, [19]) where f ðτÞ is a function of τ and some square matrices so that this integral converges.