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Analytical Properties of the Generalized Heat Matrix Polynomials Associated with Fractional Calculus
In this paper, we introduce a matrix version of the generalized heat polynomials. Some analytic properties of the generalized heat matrix polynomials are obtained including generating matrix functions, finite sums, and Laplace integral transforms. In addition, further properties are investigated using fractional calculus operators.
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]).
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 . 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.
Here and through the work, let be the vector space of all the square matrices of order ( is the set of all positive integers) whose entries are in the set of complex numbers . For a , let be the set of all eigenvalues of which is called the spectrum of . We have which imply . Here, is called the spectral abscissa of , and the matrix is said to be positive stable if . Further, let and denote the identity and zero matrices corresponding to a square matrix of any order, respectively.
Moreover, if is a matrix in which gratifies then is invertible, its inverse coincides with . Under condition (3), we can write the following Pochhammer matrix symbol
Let . Also let and be arrays of commutative matrices and commutative matrices in , respectively, such that are invertible for and all . Then, the generalized hypergeometric matrix function can be defined by (see, e.g., [11, 39])
In particular, the hypergeometric matrix function is defined by for matrices in such that is invertible for all . Also, note that for in (9), we have the Binomial type matrix function as follows:
The Laplace transform of is defined by . provided that the improper integral exists.
Lemma 1. (see ). Let be a positive stable invertible matrix in . Then, we have
3. Generalized Heat Matrix Polynomial and Generating Matrix Functions
Definition 2. Let be a positive stable matrix in the complex space satisfying the spectral condition (3). Then, we define a generalized heat matrix polynomial of degree in the following explicit form: where is the Laguerre matrix polynomial in (8).
Remark 3. Note that and that for the scalar case , taking and the polynomial coincides with the classical scalar generalized heat polynomial, see [24, 26, 33]. Further, the ordinary heat polynomial defined in , when ;
Theorem 4. Let , , , and and be positive stable matrices in such that is invertible for all A generating matrix function of is
Proof. For convenience, suppose that the left-hand side of (13) is denoted by . According to the series expression of (11) and (7) to , we find that Upon using the relation (6), the last equality evidently leads us to the required result.
Corollary 5. For , the following generating matrix function holds true
Remark 6. The Bessel matrix function , for a positive stable matrix , 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 , the following holds true
Theorem 8. Let , , and be a positive stable matrix in such that is invertible for all The following relation holds true
Proof. Follows by induction or by the successive application of Theorem 4 when The details are omitted.
Corollary 9. For in Theorem 8, the following holds true
4. Finite Sums
Here, various finite sums of can be obtained in the following results.
Theorem 11. Let , , , and be a positive stable matrix in such that is invertible for all Then, we have
Theorem 12. Let , , , also let and be positive stable matrices in such that and are invertible for all Then, we have
5. Laplace Transforms
Here, Laplace integral transforms of the generalized heat matrix polynomials are derived as follows.
Theorem 13. Let , , , . Also, let and be a positive stable matrices in such that is invertible for all The following Laplace transform formula hold
A similar procedure yields the following Laplace integral transforms. So we prefer to omit the proofs.
Theorem 14. Let , , , . Also, let and be a positive stable matrices in such that is invertible for all The following Laplace transform formula hold
Theorem 15. Let , , , . Also, let and be a positive stable matrices in such that is invertible for all The following Laplace transform formula hold The above Theorems lead to the following special cases.
Corollary 16. For is a positive stable matrix in , and , then we have the following Laplace transforms
6. Fractional Calculus Approach
Here, we consider the Riemann. Liouville fractional integral and derivative operators and of order respectively (see, for details, ) where is a function of and some square matrices so that this integral converges. where
Theorem 17. For the generalized heat matrix polynomials of degree , the following fractional integral operator holds true:
Proof. From (29) and (11), we have Setting and after a simplification, we get whose last summation, in view of (11), is easily seen to arrive at the expression in (32). This completes the proof of Theorem 17.
Theorem 18. For the generalized heat matrix polynomials , we have the following formula
Proof. According to (15), we find that Assume that the left-hand side of (35) be denoted by . Multiply the left-hand side (35) by and applying the relation (31), we get Invoking Kummer’s matrix transformation  leads to It is easy to multiply right-hand side of (35) by and applying the fractional differentiation operator from (31), whereupon this completes the establishment of the Theorem 18.
7. Concluding Remarks
In , Rosenbloom and Widder investigated expansions of solutions of the heat equation in series of polynomial solutions Further, Haimo and Markett [33, 34] discussed the generalized heat equation where is a fixed positive number and they introduced the generalized heat polynomial solution of (39) as
Note that is the ordinary heat polynomials of even order. Also, is the Hermite polynomials of even order.
Recently, many studies and extensions of the well-known special matrix polynomials have been in a focus of increasing attention leading to new and interesting problems. In this perspective, we defined the generalized heat matrix polynomials. Also, we have given some of their main properties, namely, the generating matrix functions, finite sum formulas, Laplace integral transforms, and fractional calculus operators. Further, This study is assumed to be a generalization of the scalar cases [27, 33, 34] to the matrix setting. In addition, this approach allows to derive several new integral and differential representations that can be used in theoretical, applicable aspects like the boundary value problems and the numerical algorithms. Additional research and application on this topic is now under preparation and will be presented in forthcoming works.
No data were used to support the study.
Conflicts of Interest
This work does not have any conflicts of interest.
The first-named author extends their appreciation to the Deanship of Scientific Research at King Khalid University for funding work through research groups program under grant (R.G.P.1/15/42).
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