Computation of Fourier transform representations involving the generalized Bessel matrix polynomials

Motivated by the recent studies and developments of the integral transforms with various special matrix functions, including the matrix orthogonal polynomials as kernels, in this article we derive the formulas for Fourier cosine and sine transforms of matrix functions involving generalized Bessel matrix polynomials. With the help of these transforms several results are obtained, which are extensions of the corresponding results in the standard cases. The results given here are of general character and can yield a number of (known and new) results in modern integral transforms.


Introduction
In the past few decades, the orthogonal matrix polynomials have attracted a lot of research interest due to their close relations and various applications in many areas of mathematics, engineering, probability theory, graph theory, and physics; for example, see [1][2][3][4][5][6][7][8][9]. In [4], extensions to the matrix framework of the classical families of Legendre, Laguerre, Jacobi, Chebyshev, Gegenbauer, and Hermite polynomials have been introduced. Meanwhile, one particular orthogonal polynomial family which frequently appears in the recent studies and applications [10][11][12] is that of generalized Bessel polynomials, which in its matrix form is also defined in [4,13]. Later on, distinct works on the generalized Bessel matrix polynomials have been discussed (see [14][15][16][17]).
Fourier transform (FT) is an integral transform that is used in solving different problems in mathematical physics, applied statistics, and engineering (see, [31,32]). The idea of Fourier transform is a natural extension of the idea of Fourier series. In particular, Fourier transform can accommodate aperiodic functions, which Fourier series cannot do. Recently, many results on Fourier transform and its applications have been contributed by Nicola and Trapasso [33], Urieles et al. [34], Ghodadra and Fülöp [35], Bergold and Lasser [36], and Al-Lail and Qadir [37].
On the contrary, matrix Fourier expansions and Fourier series in orthonormal matrix polynomials have been introduced by B. Osihnker in [38,39]. Defez and Jóbdar [40,41] introduced basic properties of matrix Fourier series and Fourier approximation for functions of matrix argument. Recently, Groenevelt and Koelink [42] discussed the generalized Fourier transform with hypergeometric function and matrix-valued orthogonal polynomials as kernels. Also, applications of matrix summability to Fourier transforms were established by Ş. Yildiz [43].
Motivated by some of these aforementioned investigations of the Fourier transforms of matrix-valued orthogonal polynomials, in our investigation here we study the Fouriertype transforms of the generalized Bessel matrix polynomials Y n (ξ ; F, L), ξ ∈ C, for (square) matrix parameters F and L. In particular, we obtain several Fourier cosine and sine transforms of functions involving generalized Bessel matrix polynomials with powers of the matrix, as well as matrix exponential, trigonometric, binomial, and Bessel functions. Moreover, pertinent integral transforms of the different results given here, including simpler and earlier ones, are also investigated.

Auxiliary toolbox
In this section, we recall some definitions, lemmas, and terminology which will be used to prove the main results.
Let C and N denote the sets of complex numbers and positive integers, respectively, and N 0 = N ∪ {0}. Let C n denote the n-dimensional complex vector space and C n×n denote the space of all square matrices with n rows and n columns whose entries are complex numbers.

Definition 2.1 ([4]
) For a matrix F in C n×n , the spectrum σ (F) is the set of all eigenvalues of F for which we denote where α(F) refers to the spectral abscissa of F and for which α(F) = -α(-F). A matrix F is said to be positive stable if and only if α(F) > 0.
Remark 2.1 If F, L ∈ C 1×1 = C, then the identities in Definition 2.2 reduce to those in the scalar setting.
where I is the identity matrix in C n×n .
Definition 2.4 ([4, 45]) The reciprocal gamma function denoted by -1 (w) = 1 (w) is an entire function of the complex variable ξ . Then the image of -1 (w) acting on F ∈ C n×n denoted by -1 (F) is a well-defined matrix and invertible, as well as By applying the matrix functional calculus to F, which is a positive stable matrix in C n×n , the Pochhammer symbol of a matrix argument defined by Note that, if F = -sI, where s is a positive integer, then (F) n = 0, whenever n > s. Now, from properties of the gamma matrix function, we give some lemmas which will be needed in the proof of some theorems. Lemma 2.1 Let S be a matrix in C n×n such that α(S) > 0 and w ∈ C with Re(w) > 0. The following integral formulas hold: and We thus observe that Putting S = I -R ∈ C n×n in (8) and (9), we get Similarly, we can present the following lemma.
Lemma 2.2 Let S be a matrix in C n×n such that α(S) > 0, λ, w ∈ C with Re(λ) > 0 and Re(w) > 0. The following integral formulas hold: Definition 2.5 ([4, 46]) Let k and r be finite positive integers. The generalized hypergeometric matrix function is defined by the matrix power series where F = F i , 1 ≤ i ≤ k, and L = L j , 1 ≤ j ≤ r, are commutative matrices in C n×n with L j + mI being invertible for all integers m ∈ N 0 .
Note that for k = 1, r = 0, we have the binomial-type matrix function 1 H 0 (F 1 ; -; w), |w| < 1, as follows: Also, note that for k = 2, r = 1, we get the Gauss hypergeometric matrix function 2 H 1 in the form Several special matrix functions, including the matrix orthogonal polynomials, are also presented in terms of the generalized hypergeometric matrix function in [4,46]. 4,13,16]) Let F and L be commuting matrices in C n×n such that L is an invertible matrix. For any natural number n ∈ N 0 and ξ ∈ C, the nth generalized Bessel Remark 2.2 If the matrices F, L ∈ C 1×1 = C, then the generalized Bessel matrix polynomial in (15) reduces to generalized Bessel polynomials in [10][11][12]. 47,48]) Let a matrix F ∈ C n×n satisfy the condition: β is not a negative integer for every β ∈ σ (F), then Bessel matrix function J F (w) of the first kind associated to F is given by and the modified Bessel matrix functions I F (w) and K F (w) are respectively defined as and Definition 2.8 ( [31,32]) Let f (ξ ) be a function of ξ specified for ξ > 0. Then the complex Fourier transform of f (ξ ) associated with complex frequency w is defined by together with the requirement of |F (w)| < ∞. Similarly, the inverse Fourier transform, denoted by The cosine and sine transformations, respectively, are defined similarly as follows: and Note that if f (ξ ) is an even function, then The following lemma will be required in the proof of our theorems.

Lemma 2.3 ([18]) From the basic formulae of the Fourier cosine transform
Also, if where S is a positive stable matrix in C n×n , w, λ ∈ C with Re(w) > 0, Re(λ) > 0, and K S (w) is the modified Bessel matrix function in (19).
Remark 2.3 Physically, the Fourier transform F(w) can be interpreted as an integral superposition of an infinite number of sinusoidal oscillations with different wavenumbers w (or different wavelengths τ = 2π w ). Thus, the definition of the Fourier transform is restricted to absolutely integrable functions. This restriction is too strong for many physical applications (see [31,32]).

Statement and proof of main theorems
In this section, we investigate several new interesting Fourier cosine and sine transforms of functions involving generalized Bessel matrix polynomials asserted in the following theorems: Theorem 3.1 Let S, F and L be commuting matrices in C n×n , and let Y n (λξ ; F, L) be given in (15). For the function we have where w, λ ∈ C are such that Re(w) > 0, Re(λ) > 0, and α(S) > -1.
Proof To describe the relation in (28), the proof is easy, using the well-known identities in (2). In a similar way, we can get the result in (29).
Proof The proofs of the two results in (31) and (32) can be obtained by the use of the two formulas in (10) and (11) with Definition 2.6.
Likewise, we can get the result in (35) by using (13).
After simplification, we obtain the desired result This completes the proof of Theorem 3.6.
Proof To demonstrate the truth of these results, making use of (22) with (42), we observe that The above equation gives the proof of (43).
In a similar way and by using (23) with (42), we can get the result in (44). Hence, the demonstration of Theorem 3.7 is finished.

Theorem 3.8 Let S, F, and L be commuting matrices in
then, we have where w, μ, λ ∈ C are such that Re(w) > 0, Re(μ) > 0, Re(λ) > 0, and S is a positive stable matrix in C n×n such that -1 < α(S) < 0, J S (x) is the Bessel matrix function defined in (17) and K S (x) is the modified Bessel matrix function defined in (19).
Proof The proof of this result indeed follows from applying (23) on (45), we have Using the Fourier sine transform (see [18, p. 426]), we obtain Similarly, we can arrive at the following result.
Theorem 3.9 Let Y n (ξ 2 ; F, L) be given in (15). For the function we have where w, μ, λ ∈ C are such that Re(w) > 0, Re(μ) > 0, Re(λ) > 0, Re(w) > Re(λ), S is a positive stable matrix in C n×n such that α(I -S) > 0, and S, F, and L are commuting matrices in C n×n .
Theorem 3.10 Let Y n (ξ 2 ; F, L) be given in (15). For the function we have where w, λ ∈ C are such that Re(w) > 0, Re(λ) > 0, S is a positive stable matrix in C n×n such that α(S) > -1 2 , K S (x) is modified Bessel matrix function defined in (19), and S, F, and L are commuting matrices in C n×n .
Proof In order to establish the result (50), with the help of the Lemma 2.3, we get (-nI) r F + (n -1)I r (-L) -r r!
This completes the proof.
Similarly, we can arrive at the following result.

Theorem 3.11
Let Y n (ξ 2 ; F, L) be given in (15). For the function we have where w, λ ∈ C are such that Re(w) > 0, Re(λ) > 0, and K n (w) is the modified Bessel function defined in [18].

Conclusion
In this manuscript, motivated by the recent studies and developments of the integral transforms with various special matrix functions, including the matrix orthogonal polynomials as kernels and their applications [14,17,49,50], we introduce some Fourier cosine and sine transforms of generalized Bessel matrix polynomials, together with certain elementary matrix functions, as well as exponential and Bessel functions. It is obvious that the results presented here, which involve certain matrices in C n×n , reduce to the corresponding scalar ones when n = 1.
In fact, a remarkably large number of Fourier cosine and sine formulas involving a variety of functions have been published (see, e.g., [51, pp. 7-108]). In this connection, we conclude this manuscript by posing the following problem for further investigation.
Open problem. Try to give matrix versions of the results for Fourier integral transforms (or formulas) involving a variety of special functions (see, e.g., [51, pp. 7-108]).