Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function

Umbreen Ayub; Shahid Mubeen; Amir Abbas; Aziz Khan; Thabet Abdeljawad

doi:10.1515/phys-2024-0005

Article Open Access

Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function

Umbreen Ayub , Shahid Mubeen , Amir Abbas , Aziz Khan and Thabet Abdeljawad

Published/Copyright: April 3, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

There are several problems in physics, such as kinetic energy equation, wave equation, anomalous diffusion process, and viscoelasticity that are described well in the fractional differential equation form. Therefore, the solutions with elementary solution method cannot be solved and described deliberately with detailed physics of the problems, so these problems are solved with the help of special operators such as Mittag–Leffler (M–L) functions equipped with Riemann–Liouville (R–L) fractional operators. Hence, keeping in view the above-mentioned problems in physics in the current study, the generalized properties are derived M–L functions connected with R–L fractional operators that are investigated in the generalized form. These extended special operators will be used for the solutions of generalized kinetic energy equation. The M–L function is a fundamental special function with a wide range of applications in mathematics, physics, engineering, and various scientific disciplines. Ayub et al. gave the definition of newly extended M–L ( p , s , k ) function. Also, they gave its convergence condition and found several results relevant to that. The purpose of this study is to investigate newly extended M–L function and study its elementary properties and integral transforms such as Whittaker transform and fractional Fourier transform. The R–L fractional operator is a fundamental concept in fractional calculus, a branch of mathematics that generalizes differentiation and integration to non-integer orders. In this study, we discuss the relation of M–L ( p , s , k ) -function and R–L fractional operators. In some cases, fractional calculus is used to describe kinetic energy equations, particularly in systems where fractional derivatives are more appropriate than classical integer-order derivatives. The M–L function can appear as a solution or as a part of the solution to these fractional kinetic energy equations. Also, we gave the generalization of kinetic energy equation and its solution in terms of newly extended M–L function.

Keywords: M–L (p, s, k)-function; Whittaker transform; three-parameter gamma function

1 Introduction

Mathematical analysis plays an important role in social sciences and many real-life problems. Function is the basic tool of this analysis and introduced by Gottfried Wilhem Leibniz (1646–1716). Many new functions and their applications have been discussed in the past 30 years.

Special functions play a significant role in various branches of mathematics, physics, engineering, and other fields. They are called “special” because they do not have simple or elementary representations in terms of basic functions such as polynomials, trigonometric functions, or exponentials.

Instead, they are typically defined as solutions to specific differential equations or integrals and exhibit unique properties that make them valuable in solving complex problems. Special functions have application in both field of mathematics. In applied mathematics, special functions play a vital role in solving a problem rising in the field of hydrodynamics, control theory, classical mechanics, and solution of wave equations [1–3]. Special functions often arise as solutions to differential equations that cannot be solved using elementary functions. For example, Bessel functions, Legendre polynomials, and Hermite polynomials are used to solve various types of differential equations, including those in quantum mechanics, electrodynamics, and heat conduction. Special functions, such as the Mittag–Leffler (M–L) function and the Riemann–Liouville (R–L) fractional operator, are central in fractional calculus. They are applied to model systems with memory and non-Markovian behavior, which are prevalent in viscoelastic materials, anomalous diffusion, and fractional-order control systems.

Special functions, such as the M–L function and the R–L fractional operator, are central in fractional calculus. The R–L fractional operator is a fundamental tool in fractional calculus and has numerous applications in physics, engineering, biology, and other fields where systems exhibit memory and complex behavior. Fluid dynamics and heat transfer use special functions such as the Airy function and the Whittaker function to describe fluid flow, heat conduction, and heat transfer in complex geometries. Special functions have numerous applications in mathematics, physics, engineering, and other scientific fields. These functions are essential for resolving complex problems that cannot be solved using elementary functions alone. It allows researchers and engineers to develop more accurate models and control strategies for systems that cannot be adequately described using traditional integer-order calculus.

In fluid dynamics and heat transfer, special functions such as the Airy function and the Whittaker function are used to describe fluid flow, heat conduction, and heat transfer in complex geometries [4–14].

M–L function was introduced by Gösta M–L in previous studies [15–17] in the form of summation of divergent series. After this, Wiman [18,19], Agarwal [20], and Humbert [21] discussed the M–L function in their articles. The basic properties of M–L functions such as recurrence relation, transform, derivative, and its integral representations were described in the study by Saxena [22]. The M–L function arises naturally in the solution of fractional-order integral equations or fractional-order differential equations, solution of some boundary value problem, solution of kinetic equation, and in integral operators as a kernel [22–26].

The M–L function is used to describe processes in biophysics, such as the behavior of ion channels, neuronal signaling, and the dynamics of cellular and molecular processes, which frequently exhibit non-exponential properties. The M–L function is a useful tool for solving fractional differential equations that arise in mathematical physics. It refers to time-dependent processes in physical systems. Fractional calculus is the branch of mathematical analysis, which comes in existence for operators of differentiations and integration of non-negative integer order [27,28]. Many researchers [29–37] have worked on the properties and generalizations of M–L function and discussed relations connecting the M–L functions and fractional calculus operators.

2 Preliminaries

This section contains some basic terminology and mathematical foundations, which are used in later studies.

Definition 1

In the study by Mittag–Leffler [15], the M–L function is

(2.1) E ψ ( z ) = ∑ n = 0 ∞ z n Γ ( ψ n + 1 ) ( z ∈ C , ℜ ( ψ ) > 0 ) ,

where gamma function Γ ( v ) for ℜ ( v ) > 0 is defined as

(2.2) Γ ( v ) = ∫ 0 ∞ t v − 1 e − t d t .

Definition 2

The definition of beta function is given for ℜ ( x ) > 0 and ℜ ( y ) > 0 , as

(2.3) B ( x , y ) = ∫ 0 1 u x − 1 ( 1 − u ) y − 1 d u = Γ ( x ) Γ ( y ) Γ ( x + y ) ,

(2.4) B ( x , y ) = Γ ( x ) Γ ( y ) Γ ( x + y ) .

Definition 3

A generalization of the M–L function E ψ ( z ) (2.1) by Wiman [18] is presented as

(2.5) E ψ , ϕ ( z ) = ∑ n = 0 ∞ z n Γ ( ψ n + ϕ ) ( z , ϕ ∈ C , ℜ ( ψ ) > 0 ) .

Definition 4

Prabhakar [28] introduced the three-parameter M–L function given as

(2.6) E ψ , ϕ χ ( z ) = ∑ n = 0 ∞ z n ( χ ) n Γ ( ψ n + ϕ ) n ! ( z , ϕ , χ ∈ C , ℜ ( ψ ) > 0 ) ,

where ( χ ) n is the well-known Pochhammer’s symbol defined by for χ ∈ C as

( χ ) n = χ ( χ + 1 ) ( χ + 2 ) ⋯ ( χ + n − 1 ) , for n ∈ N = 1 , for n = 0 .

Definition 5

Dorrego and Cerutti [38] defined the k -M–L function as

(2.7) E k , ψ , ϕ χ ( z ) = ∑ n = 0 ∞ ( χ ) n , k z n Γ k ( ψ n + ϕ ) n ! .

k ∈ R , z , ψ , ϕ , χ ∈ C , ℜ ( ψ ) , ℜ ( ϕ ) > 0 , and ( χ ) n , k ( z ) , where ( χ ) n , k is the Pochhammer k -symbol [39] defined as

(2.8) ( ς ) n , k = ς ( ς + k ) ( ς + 2 k ) , … , ( ς + n k ) ,

ς ∈ C , k > 0 and n ∈ N , and Γ k ( z ) is gamma k -function [39] defined as

(2.9) Γ k ( z ) = ∫ 0 ∞ t z − 1 e − t z − 1 k d t ,

where z ∈ C ∕ k Z − and k ∈ R + , ℜ ( z ) > 0 .

Definition 6

Cerutti et al. [40] presented the extended M–L ( p , k ) -function as

(2.10) E k , ψ , ϕ χ p ( z ) = ∑ n = 0 ∞ ( χ ) n , k p z n Γ k p ( ψ n + ϕ ) n ! ,

where for p , k ∈ R , z , ψ , ϕ , χ ∈ C ℜ ( ψ ) > 0 , ℜ ( φ ) > 0 , p ( χ ) n , k is the Pochhammer ( p , k ) -symbol given by Gehlot [41] as

(2.11) ( ς ) n , k p = ς p k ς p k + p … … ς p k + ( n − 1 ) p .

The connection between the three extensions of Pochhammer’s symbol is shown as [15]

(2.12) ( ς ) n , k p = p k n ( ς ) n , k = p n ς k n .

In terms of limit, the gamma ( p , k ) -function [41] is given by

(2.13) Γ k p ( ς ) = 1 k lim n → ∞ n ! p n + 1 ( n p ) ς k ( ς ) n + 1 , k p .

The gamma ( p , k ) -function Γ k p ( ς ) satisfies the following relation:

(2.14) ( ς ) n , k p = Γ k p ( ς + n k ) Γ k p ( ς ) ,

where

Γ k p ( ς + k ) = ς p k Γ k ( ς ) .

The following forms can also denote the gamma ( p , k ) -function

(2.15) Γ k p ( ς ) = ∫ 0 ∞ e − t k p t x − 1 d t .

and

(2.16) Γ k p ( ς ) = p k ς k Γ k ( ς ) = p ς k k Γ ς k .

Definition 7

Ayub et al. [42] defined a new extension of the M–L function. Let p , k ∈ R , α , β , γ ∈ C , ℜ ( α ) > 0 , ℜ ( β ) , ℜ ( γ ) > 0 , then the M–L ( p , s , k ) -function is defined as

(2.17) E k , ψ , ϕ χ , s p ( z ) = ∑ n = 0 ∞ [ χ ] n , k , s p Γ s , k p ( ψ n + ϕ ) n ! ,

where the Pochhammer ( p , s , k ) -symbol [ χ ] n , k , s p in [43] is defined as

(2.18) [ ς ] n , k , s p = ς p k s ς p k + p s … ς p k + ( n − 1 ) p s = ∏ i = 0 n − 1 ς p k + i p s ,

where

[ ς ] s = 1 − s ς 1 − s ∀ ς ∈ R .

Definition 8

Another extension of gamma function [43] denoted by Γ s , k p ( ς ) is defined as

(2.19) Γ s , k p ( ς ) = p k s ( s p ; s p ) n − 1 ( ( 1 − s ) n − 1 ) , t > 0 .

Definition 9

The relationship between [ ς ] n , k , s p and Γ s , k p ( ς ) is provided by [43]

(2.20) [ ς ] n , k , s p = Γ s , k p ( ς + n k ) Γ s , k p ( ς ) .

Definition 10

In the study by Ayub et al. [42], the relation between Γ s , k p ( ς ) , Γ k p ( ς ) , and Γ ς k is described as

(2.21) Γ s , k p ( ς ) = s ς k Γ k p ( ς ) = s p k ς k Γ k ( ς ) = ( s p ) ς k k Γ ς k ,

and Γ s , k p ( x ) is defined in equations (2.18) and (2.19), respectively. Let [ x , y ] ⊂ R be ( − ∞ < x < y < ∞ ) . The R–L fractional integrals I x + λ f and I y − λ f of order λ ∈ R , with n − 1 < λ ≤ n , n ∈ N are

(2.22) ( I x + λ f ) ( q ) = 1 Γ ( λ ) ∫ x z f ( t ) ( q − t ) 1 − λ d t ,

( q > a ; λ > 0 ) , and

(2.23) ( I b − λ f ) ( q ) = 1 Γ ( λ ) ∫ q y f ( t ) ( t − q ) 1 − λ d t ,

( q < y ; λ > 0 ) , respectively.

Definition 11

Let λ ∈ R + and n ∈ N such that n − 1 < λ < n , f ∈ L 1 ( [ 0 , ∞ ) ) . The R–L k -fractional integral of f introduced by Mubeen and Habibulah in [44] is

(2.24) ( I k λ f ) ( x ) = 1 k Γ k ( λ ) ∫ 0 t ( t − τ ) λ k − 1 f ( τ ) d τ t > 0 .

Definition 12

In the study by Wittaker and Watson [45], the Whittaker transform of f ( x ) is given as

(2.25) W f ( x ) ( t ) = ∫ 0 ∞ t ρ − 1 e − ρ t 2 W λ , μ ( ρ t ) f ( ρ t ) d t .

Definition 13

Wright hypergeometric function is defined as (see [46])

(2.26) Ψ q p ( z ) = ∑ n = 0 ∞ ∏ i = 0 p Γ ( a i + A i n ) z n ∏ j = 1 q Γ ( b j + B j n ) n ! .

Definition 14

Fox H-function is defined as

(2.27) H p , q m , n = z ( a 1 , A 1 ) … … ( a p , A p ) ( b 1 , B 1 ) … … ( b 1 , B 1 ) = 1 2 π ι ∫ L ∏ j = 1 m Γ ( b j + B j s ) ∏ j = 1 n Γ ( 1 − a j − A j s ) ∏ j = m + 1 q Γ ( 1 − b j − B j s ) ∏ j = n + 1 q Γ ( a j + A j s ) × z − s d s ,

where L is a certain contour separating the poles of the two factors in the numerator.

3 Elementary properties of newly extended M–L function

Theorem 3.1

Let E k , ψ , ϕ χ , s p ( z ) be the newly extended M–L function given by (2.17), then the following holds:

(3.1) E k , ψ , ϕ χ , s p ( z ) = 1 Γ s , k p ( ϕ ) + z ∑ n = 0 ∞ [ χ ] n + 1 , k , s p z n Γ s , k p ( ψ n + ψ + ϕ ) ( n + 1 ) ! ,

(3.2) ( s p ) ϕ E k , ψ , ϕ χ + k , s p ( z ) + ( s p ) ψ z d d z E k , ψ , ϕ + k χ , s p ( z ) = k E k , ψ , ϕ χ , s p ( z ) ,

and

(3.3) d m d x m z ϕ k − 1 E k , ψ , ϕ χ , s p z ψ k = ( s p ) − m z ϕ k − m − 1 E k , ψ , ϕ χ , s p z ψ k .

Proof

From (3.1), we have

E k , ψ , ϕ χ , s p ( z ) = ∑ n = 0 ∞ [ χ ] n , k , s p Γ s , k p ( ψ n + ϕ ) n ! , = [ χ ] 0 , k , s p Γ s , k p ( ψ ( 0 ) + ϕ ) 0 ! + ∑ n = 1 ∞ [ χ ] n , k , s p Γ s , k p ( ψ n + ϕ ) n ! , = 1 Γ s , k p ( ϕ ) + ∑ n = 0 ∞ [ χ ] n + 1 , k , s p z n + 1 Γ s , k p ( ψ ( n + 1 ) + ϕ ) ( n + 1 ) ! .

Consider the left-hand side (LHS) of (3.2), we have

(3.4) ( s p ) ϕ E k , ψ , ϕ χ + k , s p ( z ) + ( s p ) ψ z d d z E k , ψ , ϕ + k χ , s p ( z ) = ( s p ) ϕ ∑ n = 0 ∞ [ χ ] n , k , s p Γ s , k p ( ψ n + ϕ + k ) n ! + ( s p ) ψ ∑ n = 0 ∞ [ χ ] n , k , s p Γ s , k p ( ψ n + ϕ + k ) n ! , = ( s p ) ∑ n = 0 ∞ [ χ ] n , k , s p z n Γ s , k p ( ψ n + ϕ + k ) n ! ψ n + ϕ n ! .

Now, from Relation (2.21), we have

(3.5) Γ s , k p ( ψ n + ϕ ) = s ψ n + ϕ k Γ k p ( ψ n + ϕ ) , = s ψ n + ϕ k k ψ n + ϕ p Γ k p ( ψ n + ϕ + k ) , = k s p ( ψ n + ϕ ) Γ s , k p ( ψ n + ϕ + k ) , = ( s p ) ψ n + ϕ k Γ s , k p ( ψ n + ϕ + k ) .

Using (3.5) in (3.4), we obtain (3.2).

Consider the LHS of (3.3)

(3.6) d m d x m z ϕ k − 1 E k , ψ , ϕ χ , s p z ψ k = d m d x m ∑ n = 0 ∞ [ χ ] n , k , s p z ψ k n + ϕ k − 1 Γ s , k p ( ψ n + ϕ ) n ! .

Now, we consider

(3.7) d m d x m ( z ) ψ k n + ϕ k − 1 = ψ k n + ϕ k − 1 − ( m − 1 ) m ( z ) ψ k n + ϕ k − m − 1 ,

(3.8) ψ k n + ϕ k − 1 − ( m − 1 ) m = Γ ψ n + ϕ k Γ ( ψ n + ϕ k − m )

(3.9) Γ x k = k ( s p ) x k Γ s , k p ( x ) .

Substituting (3.9) into (3.8), we have

(3.10) ψ k n + ϕ k − 1 − ( m − 1 ) m = ( s p ) − m z ϕ k − m − 1 Γ s , k p ( ψ n + ϕ ) ψ n + ϕ − m k .

Substituting (3.10) into (3.6), we have

□ d m d x m z ϕ k − 1 E k , ψ , ϕ χ , s p z ψ k = ( s p ) − m z ϕ k − m − 1 ∑ n = 0 ∞ [ χ ] n , k , s p z ψ n k Γ s , k p ( ψ n + ϕ − m k ) n ! , = ( s p ) − m z ϕ k − m − 1 E k , ψ , ϕ χ , s p z ψ k .

4 Some transforms of the newly extended M–L function

In this section, we present fractional Fourier transform, Whittaker transform, and an integral expression of newly extended M–L ( p , s , k ) -function.

4.1 Whittaker transform of the newly extended M–L function

Theorem 4.1

Let p , k , c > 0 , 0 < s < 1 , ψ , ϕ , χ ∈ C with ℜ ( ψ ) > 0 , ℜ ( ϕ ) > 0 , and ℜ ( χ ) > 0 . Then, Whittaker transform of E k , ψ , ϕ χ , s p ( t ) is given by

(4.1) ∫ 0 ∞ t ρ − 1 e − ρ t 2 W λ , μ ( ρ t ) E k , ψ , ϕ χ , s p ( w t δ ) d t = k − 1 s − ψ k p − ρ − ψ k Γ χ k × Ψ 2 2 χ k , 1 , 1 2 ± m u + ρ , δ ψ k , χ k , ( 1 − λ + ρ , δ ) ; s 1 − ψ k w p 2 − ψ k ( 1 + w ) − δ ,

where Ψ 2 2 ( z ) is the Wright generalized hypergeometric function.

Proof

Let

(4.2) W = ∫ 0 ∞ t ρ − 1 e − ρ t 2 W λ , μ ( ρ t ) E k , ψ , ϕ χ , s p ( w t δ ) d t .

When we put ρ t = υ in (4.2), we have

(4.3) W = ∫ 0 ∞ υ ρ ρ − 1 e − − υ 2 W λ , μ ( υ ) E k , ψ , ϕ χ , s p w υ ρ δ 1 ρ d υ , = 1 ρ ∫ 0 ∞ υ ρ ρ − 1 e − − υ 2 W λ , μ ( υ ) ∑ n = 0 ∞ [ χ ] n , k , s p w υ ρ n Γ s , k p ( ψ n + ϕ ) n ! d υ , = ∑ n = 0 ∞ [ χ ] n , k , s p w n Γ s , k p ( ψ n + ϕ ) n ! ρ − ρ − δ n ∫ 0 ∞ e − − υ 2 υ δ n + ρ − 1 W λ , μ ( υ ) d υ , = ρ − ρ ∑ n = 0 ∞ [ χ ] n , k , s p ( w ρ δ ) n Γ s , k p ( ψ n + ϕ ) n ! × Γ 1 2 + μ + δ n + ρ Γ 1 2 − μ + δ n + ρ Γ ( 1 − λ + δ n + ρ ) .

Substituting Relations (2.11) and (2.21) into (4.3), we obtain the desired result (4.1).□

4.2 Fractional Fourier transform of newly extended M–L function

Theorem 4.2

Let p , k , c > 0 , 0 < s < 1 , ψ , ϕ , χ ∈ C with ℜ ( ψ ) > 0 , ℜ ( ϕ ) > 0 , and ℜ ( χ ) > 0 . Then the fractional Fourier transform of E k , ψ , ϕ χ , s p ( t ) is given by

(4.4) ℷ α [ E k , ψ , ϕ χ , s p ( t ) ] ( w ) = k − 1 ( s p ) ϕ k Γ χ k × ∑ n = 0 ∞ ( − 1 ) − n ( s p ) 1 − α k n ( ι ) − n − 1 w − ( n + 1 ) α Γ ( ψ n + ϕ ) k .

Proof

ℷ α [ E k , ψ , ϕ χ , s p ( t ) ] ( w ) = ∫ R e ι w 1 α t E k , ψ , ϕ χ , s p ( t ) d t , = ∫ R e ι w 1 α t ∑ n = 0 ∞ [ χ ] n , k , s p ( t n ) Γ s , k p ( ψ n + ϕ ) n ! d t , = ∑ n = 0 ∞ [ χ ] n , k , s p Γ s , k p ( ψ n + ϕ ) n ! ∫ R e ι w 1 α t ( t n ) d t .

Set ι w 1 α t = − ξ , then the aforementioned equation becomes

ℷ α [ E k , ψ , ϕ χ , s p ( t ) ] ( w ) = ∑ n = 0 ∞ [ χ ] n , k , s p Γ s , k p ( ψ n + ϕ ) n ! ∫ − ∞ 0 e − ξ − ξ ι w 1 α n − d ξ ι w 1 α , = ∑ n = 0 ∞ [ χ ] n , k , s p Γ s , k p ( ψ n + ϕ ) n ! ( ι ) n + 1 ( − 1 ) n w 1 α n + 1 ∫ − 0 ∞ e − ξ ξ n d ξ , = ∑ n = 0 ∞ [ χ ] n , k , s p Γ ( n + 1 ) ( ι ) − n − 1 ( − 1 ) − n w 1 α − n − 1 Γ s , k p ( ψ n + ϕ ) n ! .

Substituting Γ ( n + 1 ) = n ! into the aforementioned, we obtain the required result (4.4).□

4.3 Integral expression of newly defined M–L function

Theorem 4.3

Let ψ , ϕ , χ be complex numbers with ℜ ( ψ ) > 0 , ℜ ( ϕ ) > 0 , ℜ ( χ ) > 0 , 0 < s < 1 , and p , k , c > 0 . Then,

(4.5) E k , ψ , ϕ χ , s p λ ( s p ) ψ − k k = ( s p ) χ − ϕ k Γ s , k p ( χ ) ∫ 0 ∞ e − t t χ k − 1 Φ λ t ; ψ k ; ϕ k d t ,

where Φ λ t ; ψ k ; ϕ k is the Wright function defined as

Φ λ t ; ψ k ; ϕ k = ∑ n = 0 ∞ λ n t n Γ ( ψ k + ϕ k ) n ! ,

ψ > − 1 and ϕ > 0 .

Proof

By considering the right-hand side (RHS) of (4.5), we have

(4.6) I = ( s p ) χ − ϕ k Γ s , k p ( χ ) ∫ 0 ∞ e − t t χ k − 1 Φ λ t ; ψ k ; ϕ k d t , = ∑ n = 0 ∞ λ n Γ s , k p ( ϱ ) n ! ∫ 0 ∞ e − t t χ k − 1 Γ ( ψ n k + ϕ k ) d t , = ∑ n = 0 ∞ λ n Γ s , k p ( ϱ ) n ! Γ ( χ k + n ) Γ ( ψ n k + ϕ k ) .

Substituting (2.21) and (2.11) into (4.6), we obtain (4.5).□

5 R–L fractional integrals of newly extended M–L function

Theorem 5.1

Let ψ , ϕ , χ , s , η > 0 and let I − η k be the right sided R–L fractional integral operator. Then,

(5.1) I − η k u − η k − ϕ k E k , ψ , ϕ χ , s p u − ψ k ( x ) = ( s p ) η k x − ϕ k . E k , ψ , η + ϕ χ , s p x − ψ k .

Proof

By the definition of I − η k , we have

I 1 = I − η k u − η k − ϕ k E k , ψ , ϕ χ , s p u − ψ k ( x ) = 1 Γ η k ∫ x ∞ u − η k − ϕ k ( u − x ) η k − 1 ∑ n = 0 ∞ [ χ ] n , k , s p u ψ n k Γ s , k p ( ψ n + ϕ ) n ! d u .

By switching summation and integration’s respective order, we obtain

(5.2) I 1 = 1 Γ η k ∑ n = 0 ∞ [ χ ] n , k , s p Γ s , k p ( ψ n + ϕ ) n ! × ∫ x ∞ u − η k − ϕ k − ψ n k ( u − x ) η k − 1 d u .

Using the integral formula,

∫ x ∞ ( t − x ) α − 1 t − c d t = x α − c B ( α , c − α ) ,

ℜ ( α ) > 0 , and ℜ ( c ) > 0 , where B ( x , y ) is the classical beta function in (5.2), and substituting c = η k + ϕ k + ψ n k and α = χ k into (5.2), we obtain

(5.3) I 1 = 1 Γ η k ∑ n = 0 ∞ [ χ ] n , k , s p Γ s , k p ( ψ n + ϕ ) n ! × x − ϕ k − ψ n k B χ k , ϕ k + ψ n k .

Substituting (2.3) and (2.21) into (5.3), we obtain the desired result (5.1).

Corollary 5.2

If we take s → 1 in (5.1), then (5.1) becomes

I − η k u − η k − ϕ k E k , ψ , ϕ χ p u − ψ k ( x ) = p η k x − ϕ k E k , ψ , η + ϕ χ p x − ψ k ,

which coincides with the result of Wiman [18].

Corollary 5.3

If we take s → 1 , p = k = 1 in (5.1), then (5.1) becomes

I − η [ u η − ϕ E ψ , ϕ χ ( u − ψ ) ] ( x ) = p η k x − ϕ k E k , ψ , η + ϕ χ p x − ψ k ,

which coincides with the result of [22].□

Theorem 5.4

Let ψ , ϕ , χ , s , η > 0 , α ∈ R , and I + η k be the left sided R–L fractional integral operator. Then,

(5.4) I 0 + η k u ϕ k − 1 E k , ψ , ϕ χ , s p α u − ψ k ( x ) = ( s p ) η k x η + ϕ k − 1 E k , ψ , η + ϕ χ , s p α x ψ k .

Proof

Consider

I 2 = I 0 + η k u ϕ k − 1 E k , ψ , ϕ χ , s p α u − ψ k ( x ) .

Using (2.22) and interchanging the order of integration and summation, we have

(5.5) = 1 Γ η k ∑ n = 0 ∞ [ χ ] n , k , s p ( α ) n Γ s , k p ( ψ n + ϕ ) n ! ∫ 0 x u ϕ k + ψ n k − 1 ( x − u ) η k − 1 d u .

Now, consider the integral 1 Γ η k ∫ 0 x u ϕ k + ψ n k − 1 ( x − u ) η k − 1 d u in the aforementioned equation (5.5). Substituting u x = τ into the integral 1 Γ η k ∫ 0 x u ϕ k + ψ n k − 1 ( x − u ) η k − 1 d u , we obtain

(5.6) 1 Γ η k ∫ 0 x u ϕ k + ψ n k − 1 ( x − u ) η k − 1 d u = 1 Γ η k ∫ 0 1 ( x τ ) ϕ k + ψ n k − 1 ( x − x τ ) η k − 1 x d τ , = ( x ) ϕ k + ψ n k + η k − 1 Γ η k ∫ 0 1 ( 1 − τ ) η k − 1 τ ϕ k + ψ n k − 1 d τ .

Using the definition of beta function,

(5.7) = ( x ) ϕ k + ψ n k + η k − 1 Γ η k Γ η k Γ ψ n + ϕ k Γ η + ψ n + ϕ k , = ( x ) ϕ k + ψ n k + η k − 1 Γ ψ n + ϕ k Γ η + ψ n + ϕ k .

Substituting (5.7) into (5.5), we have

(5.8) I 2 = ∑ n = 0 ∞ [ χ ] n , k , s p ( α ) n Γ s , k p ( ψ n + ϕ ) n ! × ( x ) ϕ k + ψ n k + η k − 1 Γ η k Γ η k Γ ψ n + ϕ k Γ ( η + ψ n + ϕ k ) .

Substituting the relation given in [21]

Γ s , k p ( ψ n + ϕ ) ( x ) = ( s p ) x k k Γ ( x ) ,

into (5.8), we obtain the desired result (5.4).□

Corollary 5.5

If we take s → 1 in Eq. (5.4), then it coincides with the formula of Wiman [18] as

I 0 + η k u ϕ k − 1 E k , ψ , ϕ χ p α u − ψ k ( x ) = p η k x η + ϕ k − 1 E k , ψ , η + ϕ χ p α x ψ k ,

Corollary 5.6

If we choose s → 1 and p = k in Eq. (5.4), then it coincides with the result of Dorrego and Cerutti [38] as

I 0 + η k u ϕ k − 1 E k , ψ , ϕ χ α u − ψ k ( x ) = k η k x η + ϕ k − 1 E k , ψ , η + ϕ χ k α x ψ k .

Corollary 5.7

By choosing s → 1 , p = k = 1 in Eq. (5.4), then it becomes the result of Haubold et al. [47] as

I 0 + η [ u ϕ − 1 E ψ , ϕ χ ( α u ψ ) ] ( x ) = x η + ϕ − 1 E ψ , η + ϕ χ ( α x ψ ) .

Lemma 5.8

For α ∈ R , then holds the following formula:

(5.9) s α x ψ k E k , ψ , ϕ χ , s p α x ψ k = E k , ψ , ϕ − ψ χ , s p α x ψ k − E k , ψ , ϕ − ψ χ , s p α x ψ k .

Proof

Consider the LHS (5.9), we have

(5.10) s α x ψ k E k , ψ , ϕ χ , s p α x ψ k = s α x ψ k ∑ n = 0 ∞ [ χ ] n , k , s p α x ψ k n Γ s , k p ( ψ n + ϕ ) n ! .

Substituting n by m − 1 into (5.10), we have

(5.11) s α x ψ k E k , ψ , ϕ χ , s p α x ψ k = s α x ψ k ∑ m = 1 ∞ [ χ ] m − 1 , k , s p α x ψ k m − 1 Γ s , k p ( ψ ( m − 1 ) + ϕ ) ( m − 1 ) ! , = ∑ m = 1 ∞ ( m s ) [ χ ] m − 1 , k , s p α x ψ k m Γ s , k p ( ψ ( m − 1 ) + ϕ ) m ! .

Substituting the recurrence relation

m s p [ χ ] m − 1 , k , s = [ χ ] m , k , s p − [ χ − k ] m , k , s p

into (5.11), we have the required result (5.9).□

Corollary 5.9

By taking s → 1 in Lemma (5.1.8), we have the result of Wiman [18] as

α x ψ k E k , ψ , ϕ χ p α x ψ k = E k , ψ , ϕ − ψ χ p α x ψ k − E k , ψ , ϕ − ψ χ − k p α x ψ k .

Corollary 5.10

By taking s → 1 , p = k = 1 in Lemma (5.1.8), it coincides with the result of Saxena and Saigo [31]

α x ψ E ψ , ϕ χ ( α x ψ ) = E ψ , ϕ − ψ χ ( α x ψ ) − E ψ , ϕ − ψ χ − k α x ψ k .

Theorem 5.11

Let ψ , ϕ , χ , s , η > 0 , α ∈ R . Then,

(5.12) I η k u ϕ k − 1 . E k , ψ , ϕ χ , s p u ψ k ( x ) = s η k − 1 p η k x η + ϕ − ψ k − 1 E k , ψ , η + ϕ − ψ χ , s p x ψ k − E k , ψ , η + ϕ χ − k , s p x ψ k .

Proof

Considering (5.4), we have

(5.13) I η k u ϕ k − 1 E k , ψ , ϕ χ , s p u − ψ k ( x ) = ( s p ) η k x η + ϕ k − 1 E k , ψ , η + ϕ χ , s p x ψ k .

Substituting Lemma (5.1.8) into (5.13), we obtain (5.12).□

Corollary 5.12

If we take s → 1 in (5.12), we have

I η k u ϕ k − 1 E k , ψ , ϕ χ p u ψ k ( x ) = p η k x η + ϕ − ψ k − 1 E k , ψ , η + ϕ − ψ χ p x ψ k − E k , ψ , η + ϕ χ − k p x ψ k ,

which is same as the result of Wiman [18].

Corollary 5.13

Choose s → 1 and p = k = 1 in 5.12, we have

I η [ u ϕ − 1 E ψ , ϕ χ ( u ψ ) ] ( x ) = p η x η + ϕ − ψ − 1 [ E ψ , η + ϕ − ψ χ ( x ψ ) − E ψ , η + ϕ χ ( x ψ ) ] ,

which is same as the result of Saxena and Saigo [31].

6 Application of newly extended M–L function

We provide the application of newly extended M–L function by generalizing the kinetic equation involving newly extended M–L function and find its solution in terms of Fox’s H-function. In the area of fractional calculus, Houbold and Mathaia presented the first generalization of the kinetic equation in the study of Saxena et al. [48] as follows:

(6.1) N ( t ) − N 0 = − c ν I ν N ( t ) ,

where I ν is the R–L fractional integral operator. Saxena and Kalla [49] considered and studied generalizations of this equation

(6.2) N ( t ) − N 0 f ( t ) = − c ν I ν N ( t ) ( ℜ ( ν ) > 0 ) ,

where N ( t ) stands for the specie’s density at time t , N 0 for the specie’s density at time t = 0 , c is a constant, and the function f is integrable over ( 0 , ∞ ) . Starting from the previous generalization, the kinetic equation was investigated in specific, special functions, and considering various functions f . Kinetic energy equation are generalized and discussed by many researchers (see, for example, [50,51,52]).

We investigate the following generalization of kinetic equation:

(6.3) N ( t ) − N 0 u ϕ k − 1 E k , ψ , ϕ χ , s p − ( c u ) ψ k = − c ψ k I k ψ N ( t ) ,

whose solution is provided by the subsequent theorem.

Theorem 6.1

Let ψ , ϕ , χ , s , η > 0 , ℜ ( ϕ ) , ℜ ( χ ) > 0 , p , k , c ∈ R + − { 0 } . Then, the solution of (6.3) is presented as

(6.4) N ( t ) = c k N 0 ∫ 0 t H 1 , 2 1 , 1 c k ψ k ( t − τ ) ψ k − k ψ , 1 0 , ψ k − k ψ , 1 τ ϕ k × E k , ψ , ϕ χ , s p − ( c τ ) ψ k d τ .

Proof

By applying the Laplace transform on (6.3) to both sides, we obtain

(6.5) ℒ { N ( t ) } − N 0 ℒ u ϕ k − 1 E k , ψ , ϕ χ , s p − ( c u ) ψ k = − c ψ k ℒ { I k ψ N ( t ) } ,

(6.6) N ( s ) − N 0 ℒ u ϕ k − 1 E k , ψ , ϕ χ , s p − ( c u ) ψ k = − c ψ k ℒ { I k ψ N ( t ) } .

Using the following result

(6.7) ℒ { I k α ( N ( t ) ) } = ℒ { N ( t ) } ( s ) ( k s ) α k ,

and Theorem ( 10 ) in [42], we obtain

N ( s ) − N 0 k ( s p s ) − ϕ k 1 ∓ p c s k s ψ k − χ k = − c ψ k N ( s ) ( k s ) χ k ,

(6.8) N ( s ) = N 0 s ψ k s ψ k + c k ψ k . k ( s p s ) − ϕ k 1 + p c s k s ψ k χ k .

By applying inverse Laplace transform both sides of (6.8), we obtain

(6.9) ℒ − 1 N ( s ) = ℒ − 1 N 0 s ψ k s ψ k + c k ψ k k ( s p s ) − ϕ k 1 + p c s k s ψ k χ k , N ( t ) = N 0 ℒ − 1 s ψ k s ψ k + c k ψ k k ( s p s ) − ϕ k 1 + p ( c s k s ψ k ) χ k .

Using the convolution theorem

ℒ − 1 { f ( s ) g ( s ) } = ∫ 0 t 0 f ( u ) g ( t − u ) d u

and

(6.10) ℒ − 1 s ψ k s ψ k + ( c k ) ψ k = c k H 1 , 2 1 , 1 c k ψ k ( t − τ ) ψ k − k ψ , 1 0 , ψ k − k ψ , 1 .

in (6.9), we have (6.4).□

7 Conclusions

This section is concentrated on the main concluding outcomes of the current entire study. The present study is carried out to investigate the development of generalized properties of M–L functions connected with R–L operators.
This section is concentrated on the main concluding outcomes of the current entire study. The present study is carried out to investigate the development of generalized properties of M–L functions connected with R–L operators. Also, some transforms and properties are extended in the generalized form of M–L function. The relation between these properties is explored in detail
The generalized M–L functions equipped with R–L operators are utilized to find the solution of the kinetic equation in its generalized form. First, the kinetic equation in fractional form is generalized, and then, currently, generalized operators connected with M–L functions are used to find the generalized solutions of the kinetic equation.
These properties will further be used to find the solutions of extended fractional forms of integral and differential equations.
We derived some transform and properties of newly extended M–L function. Also, we discussed the relations between the newly extended generalized M–L function defined in the study by Humbert [21] and R–L fractional operator.
Furthermore, we discussed some particular cases of these relations. We gave the generalization of kinetic energy involving this newly defined generalized M–L function and its solution. By using this newly extended M–L function, we can define operators involving this M–L function as kernel. Also, we can extend this M–L function by extending further parameter in it.
In this study, we defined further generalization of the M–L and proved its various basic properties. Hence, it would be of great interest that the M–L function studied in this article will be utilized to generalize such classes of fractional and differential operators.

Acknowledgments

The authors Aziz Khan and Thabet Abdeljwad would like to thank Prince Sultan University for paying the APC and the support through the TAS research lab.

Funding information: The authors Aziz Khan and Thabet Abdeljwad would like to thank Prince Sultan University for paying the APC.
Author contributions: Conceptualization: U.A., S.M., A.A.; Methodology: A.K., T.A. Software: U.A., A.A.,S.M; Formal analysis: U.A., A.A.,S.M.; Investigation: A.A., U.A. and T.A.; Writing original draft: U.A., A.A.,A.K.; Writing review and editing: U.A., A.A., A.K., S.M., T.A.; Visualization: A.A.; Supervision: S.M.; Project administration: T.A.; Funding acquisition: T.A., A.K. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

References

[1] Rainville ED. Special functions. New Yark (USA): The Macmillan Company; 1960. Search in Google Scholar

[2] Andrews GE, Askey R, Roy R. Special functions. Cambridge, England: Cambridge University Press; 1999. 10.1017/CBO9781107325937Search in Google Scholar

[3] Erdelyi A, Magnus W, Oberhettinger F, Tricomi FG. Higher transcendental functions. vol. 3. New York, NY, USA: McGraw-Hill; 1955. Search in Google Scholar

[4] Seadawy AR. Stability analysis for Zakharov-Kuznetsov equation of weakly nonlinear ion-acoustic waves in a plasma. Comput Math Appl. 2014;61(1):172–80. 10.1016/j.camwa.2013.11.001Search in Google Scholar

[5] Seadawy AR, Iqbal M, Lu D. Applications of propagation of long-wave with dissipation and dispersion in nonlinear media via solitary wave solutions of generalized Kadomtsev-Petviashvili modified equal width dynamical equation. Comput Math Appl. 2019;78(11):3620–32. 10.1016/j.camwa.2019.06.013Search in Google Scholar

[6] Younas U, Seadawy AR, Younis M, Rizvi STR. Optical solitons and closed form solutions to the (3+1)-dimensional resonant Schrodinger dynamical wave equation. Int J Modern Phys. 2020;34(30):2050291. 10.1142/S0217979220502914Search in Google Scholar

[7] Wanga J, Shehzada K, Seadawyc AR, Arshadb M, Asmate F. Dynamic study of multi-peak solitons and other wave solutions of new coupled KdV and new coupled Zakharov-Kuznetsov systems with their stability. J Taibah Univ Sci. 2023;17(1):216389. 10.1080/16583655.2022.2163872Search in Google Scholar

[8] Seadawy AR, Rizvi STR, Ahmad S, Younis M, Baleanu D. Lump, lump-one stripe, multiwave and breather solutions for the Hunter-Saxton equation. Open Phys. 2023;19(1):1–10. 10.1515/phys-2020-0224Search in Google Scholar

[9] Selvam AGM, Baleanu D, Alzabut J, Vignesh D, Abbas S. Study of multiple lump and rogue waves to the generalized unstable space time fractional nonlinear Schrodinger equation. Adv Differ Equ. 2020;151:456.10.1186/s13662-020-02920-6Search in Google Scholar

[10] Zada A, Waheed H, Alzabut J, Wang X. Existence and stability of impulsive coupled system of fractional integrodifferential equations. J Demonstr Math. 2019;52:296–335. 10.1515/dema-2019-0035Search in Google Scholar

[11] Rizvi STR, Seadawy AR, Ahmad S, Younis M, Ali K. Study of multiple lump and rogue waves to the generalized unstable space time fractional nonlinear Schrodinger equation. Chaos Solitons Fractals. 2021;151:11125. 10.1016/j.chaos.2021.111251Search in Google Scholar

[12] Thaiprayoon C, Sudsutad W, Alzabut J, Etemad S, Rezapour S. On the qualitative analysis of the fractional boundary value problem describing thermostat control model via ψ-Hilfer fractional operator. Adv Differ Equ. 2021;2021:201. 10.1186/s13662-021-03359-zSearch in Google Scholar

[13] Baitiche Z, Derbazi C, Alzabut J, Samei ME, Kaabar MKA, Siri Z. Monotone iterative method for ψ-Caputo fractional differential equation with nonlinear boundary conditions. Fractal Fract. 2021;5(1):81. 10.3390/fractalfract5030081Search in Google Scholar

[14] Alzabut J, Selvam AGM, Dhineshbabu R, Kaabar MKA. The existence, uniqueness, and stability analysis of the discrete fractional three-point boundary value problem for the elastic beam equation. Symmetry. 2021;13(5):789. 10.3390/sym13050789Search in Google Scholar

[15] Mittag-Leffler GM. Une generalisation de l integrale de Laplace-Abel. Comptes Rendus de lAcad emie des Sciences Serie II. 1903;137:537–9. Search in Google Scholar

[16] Mittag-Leffler GM. Sur la nouvelle fonction Eα(x). Comptes Rendus del Acad emiedes Sciences. 1903;137:554–8. Search in Google Scholar

[17] Mittag-Leffler GM. Sur la representation analytiqie dune fonction monogene cinquieme note. Acta Mathematica. 1905;29(1):101–81. 10.1007/BF02403200Search in Google Scholar

[18] Wiman A. Uber den fundamental satz in der theorie der funcktionen Eα(x), Acta Mathematica, 1905;2(9):191–201. 10.1007/BF02403202Search in Google Scholar

[19] Wiman A. Uber die Nullstellun der Funktionen. Acta Mathematica. 1905;29:217–34. 10.1007/BF02403204Search in Google Scholar

[20] Agarwai RP. A propos dune note de M. Pierre Humbert. Comptes Rendus de lAcademie des Sciences. 1953;236:203–2032. Search in Google Scholar

[21] Humbert P. Quelques resultants retif sa la fonction de Mittag-Leffler. Comptes Rendus de l Academie de Sciences. 1953;236:1467–8. Search in Google Scholar

[22] Saxena RK. Certain properties of generalized Mittag-Leffler function. in Proceedings of the 3rd Annual Conference of the Society for Special Functions and Their Applications. 2002. p. 77–81. Search in Google Scholar

[23] Samko SG, Kilbas AA, Marichev OI. Fractional integrals and derivatives. Theory and applications, Gordon and Breach. New York, NY, USA; 1993. Search in Google Scholar

[24] Hilfer R. Fractional dffusion based on Riemann-Liouville fractional derivatives. J Phys Chem B. 2000;104(3):9144. 10.1021/jp9936289Search in Google Scholar

[25] Hilfer R. Applications of fractional calculus in physics. World Scientific, Singapore, Journal of Applied Mathematics. 2000. 10.1142/9789812817747Search in Google Scholar

[26] Hille E, Tamarkin JD. On the theory of linear integral equations. Ann Math. 1930;31:4798. 10.2307/1968241Search in Google Scholar

[27] Kilbas H, Srivastava H, Trujillo J. Theory and application of fractional differential equations. Elsevier; 2006. 10.3182/20060719-3-PT-4902.00008Search in Google Scholar

[28] Prabhakar T. A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math J. 1971;19:171–83. Search in Google Scholar

[29] Gorenflo R, Kilbas AA, Rogosin SV. On the generalized Mittag-Leffler type function. Integral Transform Special Funct. 1998;7(3–4):215–24. 10.1080/10652469808819200Search in Google Scholar

[30] Kilbas AA, Saigo M, Saxena RK. Solution of Volterra integro-differential equations with generalized Mittag-Leffler function in the kernels. J Integral Equ Appl. 2002;14(4):377–86. 10.1216/jiea/1181074929Search in Google Scholar

[31] Saxena RK, Saigo M. Certain properties of fractional calculus operators associated with generalized Wright function. Fract Calculus Appl Anal. 2005;6:141–54. Search in Google Scholar

[32] Kiryakova VS. Special functions of fractional calculus, recent list, results, applications. in Proceedings of the 3rd IFC Workshop Fractional Differentiation Its Applications. 2008. p. 1–23. Search in Google Scholar

[33] Saxena RK, Kalla SL, Kiryakova VS. Relations connecting multiindex Mittag-Leffler functions and Riemann-Liouville fractional calculus, algebras, groups geometries. J Appl Math. 2003;20:363–85. Search in Google Scholar

[34] Haubold H, Mathai A. The fractional kinetic equation and thermonuclear functions. Astrophys Space Sci. 2000;273:53–63. Search in Google Scholar

[35] Kumar D, Purohit SD, Secer AA, Atangana A. On generalized fractional kinetic equations involving generalized Bessel function of the first kind. Math Problems Eng. 2015;2015:289387. 10.1155/2015/289387Search in Google Scholar

[36] Nisar KS, Purohit SD, Mondal R. Generalized fractional kinetic equations involving generalized Struve function of the first kind. J King Saud Univ Sci. 28 2016;2:167–71. 10.1016/j.jksus.2015.08.005Search in Google Scholar

[37] Agarwal P, Ntouyas SK, Jain S, Chand M, Singh G. Fractional kinetic equations involving generalized k-Bessel function via Sumudu transform. Alexandria Eng J. 2018;57(3):1937–42.10.1016/j.aej.2017.03.046Search in Google Scholar

[38] Dorrego G, Cerutti R. The k-Mittag-Leffler function. Int J Contemp Math Sci. 2012;7:705–16. Search in Google Scholar

[39] Diaz R, Pariguan R. On hypergeometric functions and Pochhammer k-symbol. Divulgaciones Matemticas. 2007;15:179–92. Search in Google Scholar

[40] Cerutti R, Luciano L, Dorrego G. On the p-k-Mittag-Leffler function. Appl Math Sci. 2017;11:2541–60. 10.12988/ams.2017.78261Search in Google Scholar

[41] Gehlot K. Two parameter gamma function and its properties. 2017. p. 1–7. Search in Google Scholar

[42] Ayub U, Mubeen S, Abdeljawad T, Rahman G, Nisar KS. The new Mittag-Leffler function and its applications. J Math. 2020;2020:2463782. 10.1155/2020/2463782Search in Google Scholar

[43] Gehlot K, Nantomah K. p-q-k gamma and beta functions and their properties. Int J Pure Appl Math. 2018;118:519–26. Search in Google Scholar

[44] Mubeen S, Habibullah GM. k-fractional integrals and application. Int J Contemp Math Sci. 2012;7:89–94. Search in Google Scholar

[45] Wittaker ET, Watson GN. A course of modern analysis. Cambridge: Cambridge University Press; 1962. Search in Google Scholar

[46] Sneddon IN. The use of integral transforms. New Delhi: Tata McGraw Hill; 1979. Search in Google Scholar

[47] Haubold HJ, Mathai AM, Saxena RK. Mittag-Leffler functions and their applications. J Appl Math. 2011;2011:1–51. 10.1155/2011/298628Search in Google Scholar

[48] Saxena RK, Mathai AM, Haubold HJ. On fractional kinetic equations. Astrophys Space Sci. 2002;282(1):281–7. 10.1023/A:1021175108964Search in Google Scholar

[49] Saxena RK, Kalla SL. On the solutions of certain fractional kinetic equations. Appl Math Comput. 2008;199:504–11. 10.1016/j.amc.2007.10.005Search in Google Scholar

[50] Haubold HJ, Mathai AM. The fractional kinetic equation and thermonuclear functions. Astrophys Space Sci. 2000;273:53–63. 10.1023/A:1002695807970Search in Google Scholar

[51] Saxena RK, Mathai AM, Haubold HJ. On generalized fractional kinetic equations. Phys A. 2004;344(3–4):657–64. 10.1016/j.physa.2004.06.048Search in Google Scholar

[52] Saxena RK, Mathai AM, Haubold HJ. Unified fractional kinetic equations and a fractional diffusion equation. Astrophys Space Sci. 2004;290:241–5. 10.1023/B:ASTR.0000032531.46639.a7Search in Google Scholar

Received: 2023-09-25

Revised: 2024-01-31

Accepted: 2024-03-12

Published Online: 2024-04-03

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2024-0005

Keywords for this article

M–L (p, s, k)-function; Whittaker transform; three-parameter gamma function

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