The concept of special functions (SFs) refers to a class of mathematical functions that appear in a variety of situations, most notably in differential equation, mathematical physics, chemistry, and other fields, see, for instance, Sneddon (1956)[1], Nikiforov and Uvarov (1988)[2], and Bell (2004)[3]. These functions frequently possess unique qualities, behaviors, or forms that make them very valuable in applications. Moreover, the term special functions is applied to functions that are not trivial. In some sense the polynomials are the most special of all to the extent of being boring. The next simplest is the family of exponential functions the usual real exponential, the trigonometric and hyperbolic functions, and the general complex exponential that incorpo-rates all of them and their inverse functions. These may have been supposed to be boring but, as we shall see, they are interesting. They can be taken as solutions of first-order differential equa-tions. The next lot of special functions is the solutions of the linear second-order differential equations of mathematical physics and engineering. These include the Legendre polynomials and functions, the Laguerre, the Hermite, the Chebyshev, the Jacobi,the Gegenbauer, the Bessel and related functions and some others. All of them arise as special cases of the generalized hypergeometric functions U = pFq of p numerator and q denominator parameters (see, Dwork (1990) [4]) that are solutions of the differential equation

Y d d Y d z z + αk U = z z + βk − 1 U, (1.1) dz dz dz

k=1 k=1

where

α1 α2 · · · αp ∞   (α1)m · · · (αp)m z pFq  ; z = , (1.2) (β1)m · · · (βq)m m!

β β · · · β m=0

1 2 q

and (a)m is the usual Pochhammer symbol (or the rising factorial). For p = 2 and q = 1, we get the Gauss hypergeometric function 2F1.

The other functions are those that arose in the theory of numbers. It is of interest to follow them in more detail. The first of them, the Gamma function (Euler integral of the second kind), a generalization of the factorial function defined for integers to the real domain and thence to the complex domain (see, Davis (1959) [5]):

−y κ−1

Γ(κ) := e y dy, Re(κ) > 0. (1.3)

The function Γ is closely related to the Beta function (Euler integral of the first kind)

κ1−1 κ2−1

B(κ1, κ2) = y (1 − y) dy

Γ(κ1)Γ(κ2) = Re(κ1, κ2) > 0.

Γ(κ1 + κ2)

By using the Gamma function (1.3). Swedish Mathematician Gosta Mittag-Leffler (1903) [6] introduced the following Mittag-Leffler function of one parameter

Eα(y) = , Re(α) > 0. (1.5) Γ(rα + 1)

This function is the direct generalization of the exponential function and is known as the queen of the functions of fractional calculus. For other generalizations, properties, and applications of the Mittag-Leffler functions, the reader might consult the work by Haubold et al. (2001)[7].

Further, SFs are often highly studied, and their properties such as differential (or differ-ence ) equations, asymptotic behavior, recurrence relations, integral formulas, orthogonality, Rodrigues’ formulas, and generating functions, are extensively documented. These attributes render them effective tools for swiftly addressing intricate issues.

On top of that, recently special matrix functions (SMFs) appear in connection with statistics, Lie group theory, and theoretical physics. Many authors have dealt with orthogonal matrix poly-nomials and their applications, For examples, Hermite, Laguerre, Chebyshev, Jacobi, Gegen-bauer, Bessel and Legendre matrix polynomials were introduced and established, see, tutorial survey by Abdalla (2020) [8].

Next, further details will be provided about the importance of relationships between SFs and both fractional calculus operators (FCOs) and integral transforms (ITs).

The notion “Fractional Calculus” (FC) or “Fractional Analysis” is used for the extension of the Calculus (Analysis), when the order of integration and differentiation can be an arbitrary number (fractional, irrational, complex), that is, not obligatory integer. The FC is nowadays one of the most rapidly growing subjects of mathematical analysis in spite of the fact that it is nearly 300 years old. Where the giants of mathematics, W. Leibniz (1697), L. Euler (1730), thought about the possibility to perform differentiation of non-integer order.

The real birth and far-reaching development of the FC is due to numerous attempts of math-ematicians during the nineteenth century to beginning of twentieth century. It is practically impossible to name all important contributions made in construction of early stages of building of the FC. For example, see the survey by Machado and Kiryakova (2010)[9, 10], Machado et al. (2010)[11].

In the same vein, fractional calculus operators (FCOs), which handle non-integer (fractional) ordering on functions as a type integral transform convolutions. It is an extension of classical operators, which originated and developed with the pioneering contributions of Lacroix (1819), Liouville (1832), Letnikov (1868), Riemann (1876), Hadamard (1892), Weyl (1919), Caputo (1967) and numerous others [12]. In the literature, FCOs are defined in a wide variety of ways. Two widely used operators are the Riemann-Liouville and Caputo fractional operators [13]. Nev-ertheless, among the monographs developing the theory of FCOs involving various special func-tions and presenting some applications, we have to point out works by, Oldham and Spanier (1974) [14], Lovoie (1976)[15] Kilbas (2005) [16], Kiryakova (2008)-(2020)[17, 18], Srivastava (2015)[19], Kochubei (2019)[20], Sandev (2022)[21], and Singh (2023)[22].

Let us now begin with some various representations for FCOs.

The Fourier transform, the Fourier cosine transform, and the Fourier sine transform can be ef-fectively used for solving differential and integral equations. These transforms are also use-ful in evaluating integrals involving special functions. The reader may refer, for example to, [35, 36, 37, 38].

If f is an absolutely integrable function, the Fourier transforms are defined by [35, 36]

−2πkit

F{f(t)} = F(k) = f(t)e dt, k ∈ R,

and Z

−1 2πikt

F {F(k)} = f(t) = F(k)e dk.

Further, the fractional Fourier transform (FFT) of order β; 0 < β ≤ 1, is defined as

iω β ξ

φβ(ω) = Fβ[φ](ω) = e φ(ξ)dξ, i = −1. (2.1)

The Laplace transform of f(t) is formally defined by [35, 36]

¯ −st L{f(t)} = f(s) = e f(t)dt, Re s > 0, (2.2)

The inverse Laplace transform is Z c+i∞ −1 1 st L {f¯(s)} = f(t) = e f¯(s)ds, c > 0. (2.3) 2πi c−i∞

The Laplace transform is highly efficient for solving some class of ordinary and partial differ-ential equations. A variety of extended Laplace-type transforms are introduced for example in [36, 37, 38, 39, 40].

The pathway-type transform (Pς−transform) is given in [41, 42] in the form

P [f(w), φ] = F(φ) = [1 + (ς − 1)φ] ς−1 f(w)dw ς > 1, (2.4)

with

−ϕw

lim [1 + (ς − 1)φ] ς−1 = e , (2.5)

ς→1+ lim Pς[f(w), φ] = L[f(w), φ], (2.6)

ς→1+

where (L[., .]) is the Laplace transform (2.2).

Some basic results of the Pς− transform are given in [41] as follows

ς − 1 Pς[1, φ] = , (2.7) ln[1 + (ς − 1)φ]

n oυ+1 υ ς − 1 Pς[w , φ] = Γ(υ + 1), υ ∈ C, (2.8) ln[1 + (ς − 1)φ]

h iα −α ς − 1 Pς[ 0Dw f(w), φ] = Pς[f(w), φ], Re(α) > 0, ς > 1, (2.9) ln[1 + (ς − 1)φ]

where 0Dw f(w) is given in (1.6).

More information about the Pς−transform and its applications may be found in [41, 42].

The Hankel transform (also designated as Fourier-Bessel transform) is a fundamental tool in many areas of mathematical statistics, physics, engineering, probability theory, analytic number theory, data analysis, etc (see, for instance, [41, 42, 43, 44, 45, 46]). The integral

Hν{f(t); q} = q Jν(qt) f(t) dt, q > 0, R(ν) > , (2.10)

th

defines the Hankel transform involving the ν -order Bessel function of first kind Jν(η) [45] as a kernel. Its inverse transform is

−1 ˜ ˜ f(t) = Hν {f(q)}(t) = qJν(tq)f(q)dq, t > 0. (2.11)

The Mellin integral transform is similar with the Laplace transform and Fourier transform and is widely applied in computer science and number theory due to its invariant properties [35, 36]. The Mellin transform of a suitable integrable function f(t) is defined by n o Z ∞ ˜ p−1 f(p) = M f(t) (p) = t f(t) dt, p ∈ C, (2.12)

provided that the improper integral in (2.12) exists. The inverse Mellin transform is n o Z c+i∞ −1 1 −p f(t) = M f˜(p) (t) = t f˜(p) dp, (2.13) 2πi c−i∞

where c ∈ \{p} is a constant.

Further, there are two convolution Type theorems for the Mellin transform. If M{f(t)} = f˜(p) and M{g(t)} = g ˜(p), then

t dξ M[f(t) ∗ g(t)] = M f(ξ)g = f˜(p)g ˜(p), 0 ξ ξ Z (2.14) M[f(t) ◦ g(t)] = M f(tξ)g(ξ)dξ = f˜(p)g ˜(1 − p).

The Beta transform of f(z) is defined as [48, 49]:

a−1 b−1

B{f(z) : a, b} = z (1 − z) f(z)dz. (2.15)

The matrix version of the Beta transform of f(z) is defined as

P −I Q−I

B{f(z) : P, Q} = z (1 − z) f(z)dz, (2.16)

where P and Q are positive stable matrices in the complex matrix space of common order n; C .

For f(z) = 1, we get the Beta matrix function given by Jodar and Cortés (1998) [50] as

P −I Q−I

B (P, Q) = t (1 − t) dt. (2.17)

The Whittaker transform is defined in [51] as

W(z) = (2zt) 4 W (2zt) f(t) dt, (2.18)

whereWλ,µ(t) is the Whittaker function. For λ = and µ = ± the Whittaker transform goes over into the Laplace transform.

The generalized (Wright) hypergeometric function were first studied by Virchenko et al.(2001)[56] as follows

Γ(c) X (a) Γ(b + τ n) z

(τ) n

2R (a, b; c; z) = , τ > 0, |z| < 1. (4.1) Γ(b) Γ(c + τ n) n!

The matrix version of (4.1) was introduced by Bakhet et al. (2019) [57] as follows,

Definition 4.1. Let A, B and C be positive stable matrices in C such that the matrix C + nI is invertible for every integer n ≥ 0. Then, the Wright hypergeometric matrix function (WHMF) is defined in the form:

(τ) −1

2R (A, B; C; z) :=Γ (B)Γ(C) X z (4.2) × (A)nΓ (C + τ nI)Γ(B + τ nI) ,

where τ ∈ R+ = (0, ∞) for all integer n ≥ 0, and

  A(A + I)...(A + (n − 1)I) = Γ (A)Γ(A + nI), n ≥ 1, (A)n = (4.3) I, n = 0,

is the Pochhammer symbol of a matrix A, where

−w A−I A−I

Γ(A) = e w dw, w = exp((A − I) ln w), (4.4)

is the Gamma matrix function (see [57]). For τ = 1, (4.2) reduces to the Gauss hypergeometric matrix function which is defined by Jódar and Cortés (1998) [50].

In this section, we investigate new properties of the WHMF in (4.2). Further, we introduce the Wright type hypergeometric matrix functions by using the FCOs in (1.6) and (1.7), as follows,

R(w; A, B; ν; λ) = 2F1(A, B; (ν + 1)I; λw) Γ(ν + 1)

w, ν, λ ∈ C and A, B ∈ C ,

R(w; A, B; −µ; λ) = 2F1(A, B; (1 − µ)I; λw) Γ(1 − µ)

w, µ, λ ∈ C and A, B ∈ C .

• Main Results

Theorem 4.2. [58]. Let A, B and C be positive stable matrices in C . Then for τ > 0, m > 0, the following integrals hold true:

∞ m ∞

−u X

C−(τ+1)I

exp u (A)n Γ(B + τ nI)Γ(C)

−1 −1 −1 C − (τ + n)I n (4.7) × Γ (B)Γ (C + τ nI)Γ u du

C−τ I

z (τ) = 2R (A, B; C; z), |z| < 1.

Z 1 (τ) (τ) 2R (A, B; (m + 1)I; 1)

m τ 1

z 2R (A, B; mI; z )dz =

2R (A, B; (m + 2)I; 1) − , |z | < 1.

m(m + 1)

Theorem 4.3. [58]. Let A and B be positive stable matrices in C such that B + nI is invertible for all integer n ≥ 0 and |z| < 1, then

(τ) (τ)

(s + 1) 2R (A, B; (s + 1)I; z) − 2R (A, B; (s + 2)I; z)

τ 2 d (τ) τ = } z 2R (A, B; (s + 3)I; z) + z {τ + 2(s + 1) 2 1 (4.9) (s + 2) dz (s + 2)

d (τ) (τ) × 2R (A, B; (s + 3)I; z) + s 2R (A, B; (s + 3)I; z), Re(s) > 0.

dz

Theorem 4.4. [58]. Let A and B be positive stable matrices in C with B + nI is invertible for all integer n ≥ 0 and λ, µ ∈ C such that |λw| < 1, Re(µ) < 1. Then

I R(w; A, B; ν; λ) = R(w; A, B; ν + γ; λ), (4.10)

D R(w; A, B; ν; λ) = R(w; A, B; ν − γ; λ), (4.11)

I R(w; A, B; −µ; λ) = R(w; A, B; γ − µ; λ), (4.12)

D Rw(A, B; ν; λ) = R(w; A, B; −(γ + µ); λ). (4.13)

Theorem 4.5. [58]. Let C be a positive stable matrix in C , The Laplace transform of R(w; −nI, C + (n − 1)I; ν; λ) and R(w; −nI, C + (n − 1)I; −µ; λ); n ∈ N are given as

L R(w; −nI, C + (n − 1)I; ν; λ) = Yn(C; λ, −s), (4.14)

L R(w; −nI, C + (n − 1)I; −µ; λ) = Yn(C; λ, −s), (4.15)

where Yn(C; λ, −s) is the generalized Bessel matrix polynomial [58].

The generalized Gauss and Confluent hypergeometric matrix functions are presented by Abdall et al. (2021) [60] as follows.

∗ ∗ ∗ ∗ ∗ n×n

Definition 4.6. Assume that A, B, A , B , C and C −B are positive stable matrices in C ,

∗ ∗ ∗ ∗

such that B C = C B and p be a number with Re(p) > 0. Then, the generalized Gauss hypergeometric matrix function (GGHMF) is defined by

(A,B) ∗ ∗ ∗ ∗ (A,B) ∗ ∗ ∗

F (A , B ; C ; z; p) = (A )n B (B + nI, C − B ; p) n=0 (4.16) −1 ∗ ∗ ∗ z × B (B , C − B ) ,

and the generalized confluent hypergeometric matrix function (GCHMF) in the form

(A,B) ∗ ∗ (A,B) ∗ ∗ ∗

1F (B ; C ; z; p) = B (B + nI, C − B ; p) n=0 (4.17) −1 ∗ ∗ ∗ z × B (B ; C − B ) .

Founded on the generalized Beta matrix function

∗ ∗ −p

(A,B) ∗ ∗ B −I C −I

B (B , C ; p) = t (1 − t) 1F1 A; B; dt. (4.18) 0 t(1 − t)

and the Beta matrix function B (A, B) is given in (2.17).

Next, we give some new results for the GGHMF and GCHMF by the following theorems.

(A,B) ∗ ∗ ∗

Theorem 4.7. [60]. For the GGHMF F (A , B ; C ; z; p), the following integral form holds true.

(A,B) ∗ ∗ ∗

F (A , B ; C ; z; p)

∗ ∗ ∗ ∗

−A B −I C −B −I

= (1 − tz) t (1 − t)

−1 ∗ ∗ ∗ −p × B (B , C − B )1F1 A; B; dt, t(1 − t)

where |arg(1 − z)| < π.

Theorem 4.8. [60]. For the GGHMF with |arg (1 − z)| < π, then the following transformation formula holds true

(A,B) ∗ ∗ ∗ −A (A,B) ∗ ∗ ∗ ∗

F (A , B ; C ; z; p) = (1 − z) F A , C − B ; C ; ; p .

z − 1

(A,B) ∗ ∗ ∗

Theorem 4.9. [60]. The GGHMF F (A , B ; C ; z; p) verifies the recurrence relation

2 h i ∗ ∗ ∗ d (A,B) ∗ ∗ ∗ B(B + 3I; C − B + 3I) p F (A , B + 3I; C + 6I; z; p) dp2

∗ ∗ ∗ d (A,B) ∗ ∗ ∗ −B(B + 2I; C − B + 2I)B F (A , B + 2I; C + 4I; z; p) dp

∗ ∗ ∗ d (A,B) ∗ ∗ ∗ −B(B + I; C − B + I) p F (A , B + I; C + 2I; z; p) dp

(A,B) ∗ ∗ ∗

+AF (A , B ; C ; z; p) = 0. (4.20) (A,B) ∗ ∗

Theorem 4.10. [60]. For the GCHMF 1F (B ; C ; z; p), the following recurrence relation holds true 2 h i ∗ ∗ ∗ d (A,B) ∗ ∗ B(B + 3I; C − B + 3I) p 1F (B + 3I; C + 6I; z; p) dp ∗ ∗ ∗ d (A,B) ∗ ∗ −B(B + 2I; C − B + 2I)B 1F (B + 2I; C + 4I; z; p) dp ∗ ∗ ∗ d (A,B) ∗ ∗ −B(B + I; C − B + I) p 1F (B + I; C + 2I; z; p) dp

(A,B) ∗ ∗

+A 1F (B ; C ; z; p) = 0.

Theorem 4.11. [60]. The following Beta matrix transform formula holds true:

(A,B) ∗ ∗

B F (P + Q, B ; C ; yz; p) : P, Q

(A,B) ∗ ∗

=B(P, Q)F (P, B ; C ; w; p),

∗ ∗ n×n

where A, B, B , C , P, Q, P + Q are positive stable matrices and commuting in C with (Re(p) ≥ 0, |w| < 1).

n×n x

Theorem 4.12. [60]. If Re(s) > 0, Re(p) ≥ 0, M ∈ C and | |< 1, then the Laplace transform given in (2.2), holds true:

M−I (A,B) ∗ ∗ ∗

L z F (A , B , C , xz; p)

−M (A,B) ∗ ∗ ∗ x = s Γ(M) 1F (A , M, B ; C ; ; p).

Theorem 4.13. [60]. If ρ, δ ∈ C, Re(p) ≥ 0 and | |< 1, then the following Whittaker transform formula (2.18) holds true,

−δt

ρ−1 (A,B) ∗ ∗ ∗

t e 2 W (δt) F (A , B ; C , wt; p)dt

Γ( + µ + ρ)Γ( − µ + ρ) −ρ 2 2 (4.23) =δ × Γ(1 − λ + ρ) (A,B) ∗ 1 1 ∗ ∗ w 2F (A , ( + µ + ρ)I, ( − µ + ρ)I, B ; C , (1 − λ + ρ)I; ; p).

2 2 δ Remark 4.14. The above results improve and generalize some already known results presented by Çekim (2013) [61], Agarwal (2014) [49] and Abdalla and Bakhet (2018) [62].

In 2007, Diaz and Pariguan [63] introduced the k-hypergeometric functions as

Definition 4.15. Let k ∈ R and α1, α2, y ∈ C and α3 ∈ C \ Z , then k-hypergeometric series is defined in the form

(α ; k), (α ; k) X (α ) (α ) y 1

k 1 2 1 n,k 2 n,k

2H1 ; y = . , |y| < , (4.24) (α3; k) (α3)n,k n! k

where (·)n,k is the k-Pochhammer symbol given by

  y(y + k)...(y + (n − 1)k), n ∈ N, y ∈ C (y)n,k = 1, n = 0, k ∈ R , y ∈ C \ {0}.

Recently, numerous works have been conducted for studying the k-hypergeometric functions, for example, Asad et al. (2021)[64], Abdalla and Hidan (2021)[65], Jianrong et al. (2022)[66], and Fuli et al.(2022)[67].

Particulary, Abdalla and Hidan [65] and Hidan et al. [68] introduced and studied several properties of the following (p, k)-analogues of Gauss hypergeometric function:

(α ; k), (α ; k) X (α ) (α ) w

(p,k) 1 2 1 j,k 2 j,k

2H ; w = . , (4.25) (α3; k) (α3)j,k (pj)!

which is an entire function for p > 1, where k ∈ R and α1, α2, w ∈ C and α3 ∈ C \ Z .

The purpose of the next section is to continue the investigation of a new formulae like integral transforms and fractional calculus operators on the (p, k)-analogues of Gauss hypergeometric function (4.25).

• Main Results

(p,k)

Theorem 4.16. [56]. The Laplace transform for the 2H given in (4.25) has the following form:

( " # ) δ (p,k) (α1, k)(α2, k) L ξ k H ; uξ (α3, k) " # (4.26) kΓ (δ) (p,k) (α1, k)(α2, k)(δ, k) u = 3H ; , (ks) k (α , k) ks

− u +

α1, α2, u, ξ ∈ C, α3 ∈ C \ Z , Re(α1) > 0, Re(α2) > 0, Re(s) > 0, | | < 1, k ∈ R and p ∈ N .

0 ks

Theorem 4.17. [68]. For α1, α2, w ∈ C, α3 ∈ C\Z , Re(α1) > 0, Re(α2) > 0, k ∈ R , p ∈ N and 0 < β ≤ 1, the following fractional Fourier transform (FFT) hold true:

( " # (p,k) (α1, k)(α2, k) Fβ 2H ; w (α3, k)

(α1)n,k(α2)n,k n+1

−( ) n −(n+1)

β (4.27) = (ω) (−1) (i) (α3)n,k

p+1 1 2 p+1

p (n − )(n − )..(n − )

p p p

Theorem 4.18. [68]. For α1, α2, ν, u ∈ C, α3 ∈ C \ Z , Re(α1) > 0, Re(α2) > 0, k ∈ R , p ∈ N and 0 < Re(ν) ≤ 1, we have

( (α , k)(α , k) ) δ (p,k)   D u k H ; u k 2 1   (α3, k)   (4.28) (α1, k)(α2, k)(δ + k, k) λΓ (λ) 1−ν+δ (p,k)   = u k H ; u 3 2   kΓ (1 − ν + δ) (α3, k)(1 − ν + δ, k)

Theorem 4.19. [68]. Assume that α, β, γ, δ, η, ϑ, α1, α2 ∈ C, α3 ∈ C \ Z , x > 0, k ∈ R and p ∈ N such that Re ( ) > max 0, Re(β − δ), Re(α + β + γ − η) , then we have " # !

ϑ (p,k) (α1; k), (α2; k)

α,β,γ,δ,η −1

I w k H ; w (x)

0,x 2 1

(α3; k)

k k k

ϑ Γ (ϑ)Γ (ϑ − kβ + kδ)Γ (ϑ − kα − kβ − kγ + kη)

η −α−β+η+ −1

=k x k

k k k

Γ (ϑ + kδ)Γ (ϑ − kα − kβ + kη)Γ (ϑ − kβ − kγ + kη)

(α1;k) (α2;k) (ϑ;k) (ϑ−kβ+kδ;k) (ϑ−kα−kβ−kγ+kη;k)

(p,k)

× 5H ; x .

(α3;k) (ϑ+kδ;k) (ϑ−kα−kβ+kη;k) (ϑ−kβ−kγ+kη;k)

Remark 4.20. By invoking the Definitions (1.5), (1.6), (1.7), (1.8), (1.2), and (1.10), we find various special cases from Theorem 4.19.

In this section, we introduce a generalization of the Euler’s Beta function (EBF), which we call the extended Euler’s Beta-Logarithmic function (EEBLF). Also, we discuss various properties and establish numerical comparisons between this generalization and the previous studies using MATLAB (R2018a). Furthermore, we present a new version of the Beta distribution and acquire some of its characteristics as an application in statistics.

The extended Euler’s Beta-Logarithmic function (EEBLF) is defined in [69] as

h i Z 1 1−w w u−1 v−1 ℓ EBL α, β; u, v; ℓ = α β w (1 − w) exp − dw, 0 w(1 − w)

α, β ∈ R with α ̸= β, Re(u) > 0, Re(v) > 0, and Re(ℓ) > 0 ,

Remark 4.21. We see certain particular cases of the EBL α, β; u, v; ℓ as follows:

(i) If α = β = 1, then Eq (4.29) reduces to the extended Beta function (EBF) defined by Choudhary et al. (1997) [70] in the form

u−1 v−1 −ℓ EB (u, v; ℓ) = w (1 − w) exp dw, 0 w(1 − w)

Re(u) > 0, Re(v) > 0, and Re(ℓ) > 0 .

(ii) When ℓ = 0 in (4.29), we obtain the Beta-Logarithmic function (BLF) given by Raïssouli and Chergui (2022) [71] as

1−u u δ1−1 δ2−1

BLmean (θ, ϕ; δ1, δ2) = θ ϕ u (1 − u) du, 0 (4.31) Re(δ1) > 0, Re(δ2) > 0, θ, ϕ ∈ R such that θ ̸= ϕ .

(iii) If u = v = 1, then Eq (4.29) reduces to a new extension of Lmean, which is called the extended logarithmic mean (ELmean) as

h i Z 1 1−w w ℓ ELmean α, β; ℓ = α β exp − dw, 0 w(1 − w)

α, β ∈ R with α ̸= β, and Re(ℓ) > 0 .

(vi) If we choose ℓ = 0 in (4.32), then we get the Lmean defined by

Z   , θ ̸= ϕ,

1 ln(θ)−ln(ϕ) 1−x x

Lmean θ, ϕ = θ ϕ dx = (4.33) θ, otherwise.

(v) Taking α = β = 1 and ℓ = 0 in (4.29), we recover the classical Beta function defined in (1.4).

Defez et al. (2004) [73] introduced the Jacobi matrix polynomials in the following definition.

Definition 4.29. Let E and F be positive stable matrices in C , then the Jacobi matrix polynomial (JMP)

(E,F )

Pn (z) is defined by

(E,F ) (E + I)n −nI, E + F + (n + 1)I 1 − z Pn (z) = 2F1 ; . (4.36) n! E + I 2

These polynomials are generalizations of several families of orthogonal matrix polynomials like the Legen-dre, Chebyshev and Gegenbauer (ultraspherical) matrix polynomials.

In the current section, we define and establish the 2−variable analogue of Jacobi matrix polynomials (2VA-JMP) with some properties, which have been proposed on the pattern for 2-variable Gegenbauer matrix poly-nomials by Kahmmash (2008) [72], 2-variable Laguerre matrix polynomials by Khan and Hassan (2010) [74], 2-variable Hermite generalized matrix polynomials by Subuhi et al. (2010)[75], and 2-variable Shivley’s matrix polynomials by He et al. (2019) [76].

Definition 4.30. [77]. Let E and F be the positive stable matrices in C such that E + nI and F + nI are invertible for all integers n ≥ 0. Then Jn(E, F, z, w) takes the following explicit form:

X (E + I) (F + I)

n n −1 −1

Jn(E, F, z, w) = [(E + I)s] [(F + I)n−s] s!(n − s)!

z − w s z + w n−s × ( ) ( ) ,

where E and F satisfy the conditions EF = F E, and

Re(z) > −1 ∀ z ∈ σ(E), Re(z) > −1 ∀ z ∈ σ(F), (4.38)

with (E)n denotes the matrix Pochhammer symbol.

• Main Results Theorem 4.31. [77]. Let E, F ∈ C be positive stable matrices. The generating matrix function of Jn(E, F, z, w) is

−1 ν −(I+E+F )

(E + F + I)ν [(E + I)ν ] Jν (E, F, z, w) t = (1 − t) " # (4.39) (E + F + I), (2I + E + F) (z − w) (z − w) × 2F1 ; 2t , |t| < 1, | | < 1.

I + E (1 − t) (1 − t)

d×d t t

Theorem 4.32. [77]. Let E and F be positive stable matrices in C and | (z− w)| < 1 and | (z+ w)| < 1, the following Bateman’s generating matrix function holds true:

−1 −1 ν

Jν (E, F, z, w)[(E + I)ν ] [(F + I)ν ] t " # " # (4.40) − t √ − t √ = 0F1 ; (z − w) · 0F1 ; (z + w) .

E + I 2 F + I 2

Theorem 4.33. [77].. Let D, E and F be commutative matrices in C such that E and F satisfies the

1− wt−R 1+ wt−R

spectral condition (4.38) with < 1 and < 1, the following Brafman’s generating matrix function holds true:

−1 −1 ν

(D)ν (E + F − D + I)ν [(E + I)ν ] [(F + I)ν ] Jν (E, F, z, w) t

" √ # D, E + F − D + I 1 − wt − R = 2F1 ; E + I 2 " √ # D, E + F − D + I 1 + wt − R −1 × F ; , R = (1 − 2zt + wt ) 2 .

F + I 2

Theorem 4.34. [77]. Let E and F be matrices in C satisfing (4.38). Then the 2VAJMP Jn(E, F, z, w) may be expressed as

−E −F

(z − w) (z + w) Jn(E, F, z, w) = 2 n!

" # (4.41) √ √ d

n E+nI F +nI

× D (z − w) (z + w) , D ≡ √ d(z ± w)

Theorem 4.35. [77]. The 2VAJMP Jn(E, F, z, w) satisfies the following differential matrix recurrence rela-tions:

2z(E + F + nI) DJn(E, F, z, w) h √ i + z(E − F) − (E + F + 2nI) w DJn−1(E, F, z, w)

= w(E + F + nI) 2n Jn(E, F, z, w) − (E − F) Jn−1(E, F, z, w) ,

2 w(E + F + nI) DJn(E, F, z, w) h√ i + w(E − F) − (E + F + 2nI)z DJn−1(E, F, z, w) (4.43)

= w(E + F + nI) (E + F + 2nI) Jn−1(E, F, z, w),

D Jn(E, F, z, w) = 2 (E + F + (n + 1)I)s Jn−s(E + sI, F + sI, z, w), 0 < s ≤ n.

Remark 4.36. The current results enhance and expand previous findings by Rehana et al. (2017)[78].

Later on, evaluations of Hankel transform representations involving a variety of special functions and polyno-mials have been archived by Debnath and Bhatta (2016) [79, Chapter 7]. Also, Abdalla et al. (2021) discussed various integral transforms such as Fourier transforms [80], Laplace transforms [81], and matrix Riemann– Liouville fractional integrals [82] associated with functions involving generalized Bessel matrix polynomials.

Motivated by previous work, we give a generalization of the scalar Hankel transform (2.10) and its inverse into the matrix framework.

Definition 4.37. [83].(Matrix Hankel Transforms) Let S be a positive sable matrix in C and let Φ(u) be a function defined for u ≥ 0. The Hankel transform involving Bessel matrix function as kernel of Φ(u) is defined as

ΞS(v) ≡ HS{Φ(u); v} ≡ Φ(u) uv JS(uv) du, (4.45)

where v > 0 and JS(uv) is the Bessel matrix function of the first kind defined by Jódar et al. (1994) [85].

If β e(S) > 1/2, Hankel’s repeated integral immediately gives the inversion formula

Φ(u) = HS {ΞS(v); u} ≡ ΞS(v) uv JS(uv) dv. (4.46)

• Main Results Now, we introduce the evaluation of the Hankel matrix transforms with products of certain elementary functions and the generalized Bessel matrix polynomials Bn(u; M, N)

X (−1)

−1 m

[86].Bn(u; M, N) = (−nI)m(M + (n − 1)I)m(u N ) , (4.47)

where M and N are commuting matrices in C such that N is an invertible matrix.

Theorem 4.38. [86]. Let S and N be commuting matrices in C . If

S+(2n+ )I −σu 2

Φ(u) = u 2 e B (N; (1 − 2n)I − S, u ), (4.48)

then, we have

1 −v

−(S+(2n+1)I) S+(2n+ )I

Ξ (v) = (2σ) v 2 e 4σ

×Bn(4σ(I − σN); (1 − 2n)I − S, −v ),

n×n e where S is a positive stable matrix in C , β(S + nI) > −1, v > 0 and σ ∈ C such that Re(σ) > 0.

Theorem 4.39. [86]. Let S, P and M be positive stable and commuting matrices in C . When

P + I −σu 2

Φ(u) = u 2 e B (1; M, σu ), (4.50)

then, we have

n −( (P +S)+I) −(S+I) S+ I

Ξ (v) = (−1) σ 2 (2) v 2 1 1 −1 1 −1 ×Γ( (P + S)) (M + (P + S))n Γ (S + I) [(−( (P + S)))n] 2 2 2   (4.51) (1 − n)I + (P + S), nI + M + (P + S),

2 2 2

 v  × 2F2  ; −  , S + I, M + (P + S)

where β(P) > − , v > 0 and σ ∈ C such that Re(σ) > 0.

Theorem 4.40. [86]. Let Bn(u; M, N) be given in (4.47). If

Φ(u) = Bn(σu ; M, N), (4.52) then, we have 1 1 3 1 1

−1 −1

Ξ (v) =2 2 v Γ( S + I) Γ ( S + I) 2 4 2 4

1 3 −1 3

−nI, M + (n − 1)I, S + I, S + I,

2 4 2 4

  (4.53)  2 −1 × 4F0  ; 4σ(v N)  ,

where β e(S) > −1/2 and v > 0.

Theorem 4.41. [86]. Let Bn(u; M, N) be given in (4.47). When

Φ(u) = log u Bn(λu ; M, N), (4.54)

then, we have 1 1 −1 1 ΞS(v) = √ Γ( S + 3/4I) Γ ( S + 1/4I) v 2 2 2

× (−nI)m (M + (n − 1)I)m

2 −1 m

1 −1 (4λ(N v ) ) × ( S + 3/4I)m ( S + 3/4I)m 2 2 m!

1 1 2 × ψ( S + (3/4 + m)I) + ψ( S + (1/4 − m)I) − log(v /4)

n×n e where M, N and S are commuting matrices in C such that β(S) > −3/2, ψ(S) is the digamma matrix function defined by

−1 ′

ψ(S) = Γ (S)Γ (S),

−1 ′

where Γ (S) and Γ (S) are reciprocal and derivative of the Gamma matrix function, respectively and v > 0, λ > 0.

Remark 4.42. Abdalla et al. (2021)[56] work into the Hankel matrix transform revealed further features and applications such as the convolution property and solve of partial differential equations of second order. 4.7 Applications.

Nowadays, an interesting application of special functions in allied sciences is solving generalized fractional kinetic equations by using various integral transforms, including the recent papers by Agarwal et al. (2018) [80], Singh et al. (2021) [85], Abdalla and Akel (2022) [83], Hidan et al. (2022) [89], Chand et al. (2024) [88], and Alqarni (2024) [89]. Here, we will show two applications to fractional kinetic equations.

Application of Mellin integral transform in solving generalized kinetic equations involving