On the generalized fractional integrals of the generalized Mittag-Leffler function

Article metrics

• 1024 Accesses

• 5 Citations

Abstract

Abstract

In this paper, we employ the generalized fractional calculus operators on the generalized Mittag-Leffler function. Some results associated with generalized Wright function are obtained. Recent results of Chaurasia and Pandey are obtained as special cases.

2000 Mathematics Subject Classification

33C45, 47G20, 26A33

Introduction

In 1903, the Swedish mathematician Mittag-Leffler (1903) introduced the function

$E ν (z)= ∑ s = 0 ∞ z s Γ ( νs + 1 ) ,(ν>0,z∈ℂ),$
(1.1)

where z is a complex variable and ν≥0. The Mittag-Leffler function is a direct generalization of exponential function to which it reduces for ν=1. For 0<ν<1 it interpolates between the pure exponential and hypergeometric function$1 1 - z .$ Its importance is realized during the last two decades due to its involvement in the problems of physics, chemistry, biology, engineering and applied sciences. Mittag-Leffler function naturally occurs as the solution of fractional order differential or fractional order integral equation. The generalization of E ν (z) was studied by Wiman (1905) and he defined the function as

$E ν , ρ (z)= ∑ s = 0 ∞ z s Γ ( νs + ρ ) ,(ν>0,ρ>0,z∈ℂ),$
(1.2)

which is known as Wiman function.

In 1971, Prabhakar (1971) introduced the function$E ν , ρ δ (z)$ in the form of

$E ν , ρ δ (z)= ∑ s = 0 ∞ ( δ ) s z n Γ ( νs + ρ ) s !$
(1.3)

where$ν,ρ,δ,z∈ℂ,Re(ν>0)$ and$E ν , ρ δ (z)$ is an entire function of order [ R e(ν)]-1. Special Cases: (i) Setting δ=1 in (1.3), we have

$E ν , ρ 1 ( z ) = E ν , ρ ( z )$

(ii) Setting ρ=δ=1 in (1.3), we have

$E ν , 1 1 ( z ) = E ν ( z )$

(iii) Setting ν=ρ=0 in (1.3), we have

$E 0 , 0 δ ( z ) = 1 F 0 ( δ ; z )$

For various properties and other details of (1.3), see (Kilbas et al.2004).

The generalized Wright function p Ψ q defined for$z∈ℂ, a i , b j ∈ℂ$ and α i ,β j R(α i ,β j ≠0;i=1,2,...,p; j=1,2,...,q) is given by the the series

$p Ψ q (z)= p Ψ q ( a i , α i ) ( 1 , p ) ; z ( b j , β j ) ( 1 , q ) ; = ∑ s = 0 ∞ ∏ i = 1 p Γ ( a i + α i s ) z s ∏ j = 1 q Γ ( b j + β j s ) s ! ,$
(1.4)

where Γ(z) is the Euler gamma function ((Erdlyi et al.1953), Sec. 1.1) and the function (1.4) was introduced by Wright (1935) and is known as generalized Wright function. Conditions for the existence of the generalized Wright function (1.4) together with its representation in terms of Mellin-Barnes integral and in terms of H-function were established in (Wright1934). Some particular cases of generalized Wright function (1.4) were established in ((Wright 1934), Sec. 6). Wright (1940a,c) investigated, by “steepest descent” method, the asymptotic expansions of the function ϕ(α,β;z) for large values of z in the cases α>0 and -1<α<0, respectively. In Wright (1940c) indicated the application of the obtained results to the asymptotic theory of partitions. In (Wright19351940a,b) Wright extended the last result to the generalized Wright function (1.4) and proved several theorems on the asymptotic expansion of generalized Wright function p Ψ q (z) for all values of the argument z under the condition,

$∑ j = 1 q β j - ∑ i = 1 p α i >-1.$
(1.5)

For a detailed study of various properties, generalizations and applications of Wright function and generalized Wright function, we refer to papers of Wright (193419351940a,b,c) and Kilbas (2002)

Fractional calculus operators and generalized fractional calculus operators

The left and right-sided Rimann-Liouville fractional calculus operators are defined by Samko et al. (1993), Sec. 5.1. For αC(R e(α)>0)

$I 0 + α f = 1 Γ ( α ) ∫ 0 x f ( t ) ( x - t ) 1 - α dt ,$
(2.1)
$I 0 - α f = 1 Γ ( α ) ∫ x ∞ f ( t ) ( x - t ) 1 - α dt , s$
(2.2)
$D 0 + α f ( x ) = d dx [ α ] + 1 I 0 + 1 - α f ( x ) = 1 Γ ( 1 - { α } ) d dx [ α ] + 1 ∫ 0 x f ( t ) ( x - t ) { α } dt ,$
(2.3)
$D 0 - α f ( x ) = d dx [ α ] + 1 I 0 - 1 - α f ( x ) = 1 Γ ( 1 - { α } ) d dx [ α ] + 1 ∫ 0 x f ( t ) ( x - t ) { α } dt ,$
(2.4)

where [α] means the maximal integer not exceeding α and {α} is the fractional part of α.

An interesting and useful generalizations of the Riemann-Liouville and Erdlyi-Kober fractional integral operators has been introduced by Saigo (1978) in terms of Gauss hypergeometric function as given below. Let α,β,γC and xR+, then the generalized fractional integration and fractional differentiation operators associated with Gauss hypergeometric function are defined as follows:

$I 0 + α , β , γ f ( x ) = x - α - β Γ ( α ) ∫ 0 x ( x - t ) α - 1 2 F 1 α + β , - γ ; 1 - t x f ( t ) dt ,$
(2.5)
$I 0 - α , β , γ f ( x ) = 1 Γ ( α ) ∫ x ∞ ( t - x ) α - 1 t - α - β 2 F 1 α + β , - γ ; α ; 1 - x t × f ( t ) dt ,$
(2.6)
$D 0 + α , β , γ f (x)= I 0 + - α - β , α + γ f(x)= d dx k I 0 + - α + k , - β - k f (x),$
(2.7)
$( Re ( α ) > 0 ) ; k = [ Re ( α ) + 1 ] D 0 - α , β , γ f = I 0 - - α - β , α + γ f ( x ) = - d dx k I - - α + k , - β - k , α + γ f ( x ) , ( Re ( α ) > 0 ) ; k = [ Re ( α ) + 1 ]$
(2.8)

Operators (2.5)-(2.8) reduce to that in (2.1)-(2.4) as follows:

$( I 0 + α , - α , γ f ) ( x ) = I 0 + α f ( x ) ,$
(2.9)
$( I 0 - α , - α , γ f ) ( x ) = I 0 - α f ( x ) ,$
(2.10)
$( D 0 + α , - α , γ f ) ( x ) = D 0 + α f ( x ) ,$
(2.11)
$( D 0 - α , - α , γ f ) ( x ) = D 0 - α f ( x ) .$
(2.12)

Here, we also need the basic result given below (see Rainville (1960), Theorem 18, p. 49.)

Lemma1

If Re(c-a-b)>0 and if c is neither zero nor a negative integer, then

$F ( a , b ; c ; 1 ) = Γ ( c ) Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) .$
(2.13)

Left-sided generalized fractional integration of generalized Mittag-Leffler function

In this section we consider the left-sided generalized fractional integration formula of the generalized Mittag-Leffler function.

Theorem1.

If$α,β,γ,ρ,δ∈ℂ$, R e(α)>0, R e(ρ+γ-β)>0,ν>0,λ>0, and aR. If the condition (1.5) is satisfied and$I 0 + α , β , γ$ be the left-sided operator of generalized fractional integration associated with Gauss hypergeometric function, then there holds the following formula:

$I 0 + α , β , γ ( t ρ - 1 ) E ν , ρ δ [ a t λ ] ( x ) = x ρ - β - 1 Γ ( δ ) 3 Ψ 3 × ( ρ - β + γ , λ ) , ( ρ , λ ) , ( δ , 1 ) ; ax λ ( ρ - β , λ ) , ( α + ρ + γ , λ ) , ( ρ , ν ) ; ,$
(3.1)

Proof

Denote L.H.S. of Theorem 1 by Ω, then

$Ω = I 0 + α , β , γ ( t ρ - 1 ) E ν , ρ δ [ at λ ] ( x )$

Using the definition of generalized Mittag-Leffler function (1.3) and fractional integral formula (2.5), we get

$Ω = x - α - β Γ ( α ) ∫ 0 x ( x - t ) α - 1 2 F 1 α + β , - γ ; 1 - t x × ( t ρ - 1 ) E ν , ρ δ [ a t λ ] dt$

By using Gauss hypergeometric series ((Srivastava and Karlsson1985), p.18, Eq. 17), series form of generalized Mittag-Leffler function (1.3), interchanging the order of integration and summations and evaluating the inner integral by the use of the known formula of Beta Integral and finally by the use of above lemma, we have

$Ω = x ρ - β - 1 Γ ( α ) ∑ s = 0 ∞ Γ ( ρ - β + γ + λs ) Γ ( ρ + λs ) Γ ( δ + s ) Γ ( ρ - β + λs ) Γ ( ρ + α + γ + λs ) Γ ( ρ + νs ) × ( a x λ ) s s !$

or

$Ω = x ρ - β - 1 Γ ( δ ) 3 Ψ 3 ( ρ - β + γ , λ ) , ( ρ , λ ) , ( δ , 1 ) ; a x λ ( ρ - β , λ ) , ( α + ρ + γ , λ ) , ( ρ , ν ) ; ,$

which completes the proof. □

Corollary1

If$α,β,γ,ρ,∈ℂ$, R e(α)>0, R e(ρ+γ-β)>0,ν>0,λ>0, aR and if the condition (1.5) is satisfied and$I 0 + α , β , γ$ be the left-sided operator of generalized fractional integration associated with Gauss hypergeometric function, then there holds the following formula:

$I 0 + α , β , γ ( t ρ - 1 ) E ν , ρ [ a t λ ] ( x ) = x ρ - β - 1 3 Ψ 3 × ( ρ - β + γ ) , ( ρ , λ ) , ( 1 , 1 ) ; ax λ ( ρ - β , λ ) , ( α + ρ + γ , λ ) , ( ρ , ν ) ; .$
(3.2)

Remark1

If we set λ=ν in our result (3.1), we arrive at the result ((Chaurasia and Pandey2010), (3.1)) given by Chaurasia and Pandey.

Right-sided generalized fractional integration of generalized Mittag-Leffler function

In this section we consider the left-sided generalized fractional integration formula of the generalized Mittag-Leffler function.

Theorem2.

If$α,β,γ,ρ,δ∈ℂ$, R e(α)>0, R e(α+ρ)>m a x[ -R e(β),-R e(γ)] with the conditions R e(β)≠R e(γ),ν>0,λ>0 and aR. If the condition (1.5) is satisfied and$I 0 - α , β , γ$ be the right-sided operator of generalized fractional integration associated with Gauss hypergeometric function, then there holds the following formula:

$I 0 - α , β , γ ( t - α - ρ ) E ν , ρ δ [ a t - λ ] ( x ) = x - ρ - α - β Γ ( δ ) 3 Ψ 3 × ( α + β + ρ , λ ) , ( α + ρ + γ , λ ) , ( δ , 1 ) ; a x - λ ( ρ , ν ) , ( α + ρ , λ ) , ( 2 α + β + γ + ρ , λ ) ;$
(4.1)

Proof.

Denote L.H.S. of Theorem 2 by I2, then

$I 2 = I 0 - α , β , γ ( t - α - ρ ) E ν , ρ δ [ a t - λ ] ( x )$

Using the definition of generalized Mittag-Leffler function (1.3) generalized fractional integral formula (2.6) and proceeding similarly to the proof of theorem 1, we have

$I 2 = x - ρ - α - β Γ ( δ ) ∑ s = 0 ∞ Γ ( α + β + ρ + λs ) Γ ( α + ρ + γ + λs ) Γ ( δ + s ) Γ ( α + ρ + λs ) Γ ( 2 α + β + γ + ρ + λs ) Γ ( ρ + νs ) × ( a x - λ ) s s !$

or

$I 2 = x - ρ - α - β Γ ( δ ) 3 Ψ 3 ( α + β + ρ , λ ) , ( α + ρ + γ , λ ) , ( δ , 1 ) ; a x - λ ( ρ , ν ) , ( α + ρ , λ ) , ( 2 α + β + γ + ρ , λ ) ; .$

Corollary2.

If$α,β,γ,ρ,δ∈ℂ$, R e(α)>0, R e(α+ρ)>m a x[-R e(β),-R e(γ)] with the conditions R e(β)≠R e(γ),ν>0,λ>0 and aR. If the condition (1.5) is satisfied and$I 0 - α , β , γ$ be the right-sided operator of generalized fractional integration associated with Gauss hypergeometric function, then there holds the following formula:

$I 0 - α , β , γ ( t - α - ρ ) E ν , ρ δ [ a t - λ ] ( x ) = x - ρ - α - β 3 Ψ 3 × ( α + β + ρ , λ ) , ( α + ρ + γ , λ ) , ( 1 , 1 ) ; a x - λ ( ρ , ν ) , ( α + ρ , λ ) , ( 2 α + β + γ + ρ , λ ) ; .$
(4.2)

Remark2.

If we set λ=ν in our result (4.1), we arrive at the result ((Chaurasia and Pandey2010), (4.1)) given by Chaurasia and Pandey.

Left-sided generalized fractional differentiation of generalized Mittag-Leffler function

In this section we consider the left-sided generalized fractional differentiation formula of the generalized Mittag-Leffler function.

Theorem3.

If$α,β,γ,ρ,δ∈ℂ$, R e(α)>0, R e(ρ+β+γ)>0,ν>0,λ>0, and aR. If the condition (1.5) is satisfied and$D 0 + α , β , γ$ be the left-sided operator of generalized fractional differentiation associated with Gauss hypergeometric function, then there holds the following formula:

$D 0 + α , β , γ ( t ρ - 1 ) E ν , ρ δ [ a t λ ] ( x ) = x ρ + β - 1 Γ ( δ ) 3 Ψ 3 × ( α + β + γ + ρ , λ ) , ( ρ , λ ) , ( δ , 1 ) ; a x λ ( ρ + β , λ ) , ( ρ + γ , λ ) , ( ρ , ν ) ; .$
(5.1)

Proof.

Denote L.H.S. of Theorem 3 by I3, then

$I 3 = D 0 + α , β , γ ( t ρ - 1 ) E ν , ρ δ [ a t λ ] ( x )$

Using the definition of generalized Mittag-Leffler function (1.3) and fractional differentiation formula (2.7), we have

$I 3 = d dx k I 0 + - α + k , - β , α + γ - k ( t ρ - 1 ) E ν , ρ δ [ a t λ ] = d dx k x α + β Γ ( - α + k ) ∫ 0 x ( x - t ) - α + k - 1 2 F 1 × - α - β , - γ - α + k ; - α + k ; 1 - t x . ( t ρ - 1 ) E ν , ρ δ [ a t λ ] dt I 3 = x ρ + β - 1 Γ ( δ ) ∑ s = 0 ∞ Γ ( ρ + α + β + γ + λs ) Γ ( ρ + λs ) Γ ( δ + s ) Γ ( ρ + β + λs ) Γ ( ρ + γ + λs ) Γ ( ρ + νs ) ( a x λ ) s s ! or I 3 = x ρ + β - 1 Γ ( δ ) 3 Ψ 3 ( ρ + α + β + γ , λ ) , ( ρ , λ ) , ( δ , 1 ) ; a x λ ( ρ + β , λ ) , ( ρ + γ , λ ) , ( ρ , ν ) ; .$

Corollary3.

If$α,β,γ,ρ,∈ℂ$, R e(α)>0, R e(ρ+β+γ)>0,ν>0,λ>0, and aR. If the condition (1.5) is satisfied, then there holds the following formula:

$D 0 + α , β , γ ( t ρ - 1 ) E ν , ρ δ [ a t λ ] ( x ) = x ρ + β - 1 3 Ψ 3 × ( ρ + α + β + γ , λ ) , ( ρ , λ ) , ( 1 , 1 ) ; a x λ ( ρ + β , λ ) , ( ρ + γ , λ ) , ( ρ , ν ) ; .$
(5.2)

Remark3.

If we set λ=ν in our result (5.1), we arrive at the result ((Chaurasia and Pandey2010), (5.1)) given by Chaurasia and Pandey.

Right-sided generalized fractional differentiation of generalized Mittag-Leffler function

In this section we consider the left-sided generalized fractional differentation formula of the generalized Mittag-Leffler function.

Theorem4.

If$α,β,γ,ρ,δ∈ℂ$, R e(α)>0, R e(ρ)>m a x[R e(α+β+k,-R e(γ)],ν>0,λ>0, and aR with R e(α+β+γ)+k≠0 (where k= [ R e(α)]+1). If the condition (1.5) is satisfied and$D 0 - α , β , γ$ be the right-sided operator of generalized fractional differentiation associated with Gauss hypergeometric function, then there holds the following formula:

$D 0 - α , β , γ ( t α - ρ ) E ν , ρ δ [ a t - λ ] ( x ) = x α + β - ρ Γ ( δ ) 3 Ψ 3 × ( ρ + γ , λ ) , ( ρ - α - β , λ ) , ( δ , 1 ) ; a x - λ ( ρ - α , λ ) , ( ρ + γ - α - β , λ ) , ( ρ , ν ) ; .$
(6.1)

Proof.

Denote L.H.S. of Theorem 4 by I4,then

$I 4 = D 0 - α , β , γ ( t α - ρ ) E ν , ρ δ [ a t - λ ] ( x )$

Using the definition of generalized Mittag-Leffler function (1.3) and fractional differentation formula (2.8), we have

$I 4 = - d dx k I 0 - - α + k , - β - k , α + γ ( t α - ρ ) E ν , ρ δ [ a t - λ ] = - d dx k 1 Γ ( - α + k ) ∫ x ∞ ( t - x ) - α + k - 1 t α + β 2 F 1 × - α - β , - α - γ ; - α + k ; 1 - x t . ( t α - ρ ) E ν , ρ δ [ a t - λ ] dt = x α + β - ρ Γ ( δ ) ∑ s = 0 ∞ Γ ( ρ - α - β + λs ) Γ ( ρ + γ + λs ) Γ ( δ + s ) Γ ( ρ - α - β + γ + λs ) Γ ( ρ - α + λs ) Γ ( ρ + νs ) ( a x - λ ) s s ! or I 4 = x α + β - ρ Γ ( δ ) 3 Ψ 3 ( ρ - α - β , λ ) , ( ρ + γ , λ ) , ( δ , 1 ) ; a x - λ ( ρ - α - β + γ , λ ) , ( ρ - α , λ ) , ( ρ , ν ) ; .$

Remark4

If we set λ=ν in our result (6.1), we arrive at the result ((Chaurasia and Pandey2010), (6.1)) given by Chaurasia and Pandey.

References

1. Chaurasia VBL, Pandey SC: On the fractional calculus of generalized Mittag-Leffler function. Scientia, Ser A; Math Sci 2010, 20: 113-122.

2. Erdlyi A, Magnus W, Oberhettinger F, Tricomy FG: Higher transcendental functions. McGraw-Hill, New York, Toronto, London; 1953.

3. Kilbas AA, Saigo M, Trujillo JJ: On the generalized Wright function. Frac Calc Apl Anal 2002, 5(4):437-460.

4. Kilbas AA, Saigo M, Saxena RK: Generalized Mittag-Leffler function and generalized fractional calculus operators. Integral Transforms Spec Funct 2004, 15: 31-49. 10.1080/10652460310001600717

5. Mittag-Leffler GM: Sur la nouvelle fonction E α ( x ). C R Acad Sci Paris 1903, 137: 554-558.

6. Prabhakar TR: A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math J 1971, 19: 7-15.

7. Rainville ED: Special functions. New York: Macmillan; 1960.

8. Saigo M: A remark on integral operators involving the Gauss hypergeometric functions. Math Rep Kyushu Univ 1978, 11: 135-143.

9. Samko SG, Kilbas AA, Marichev OI: Fractional integrals and derivatives: theory and applications. Yverdon, Switzerland: Gordon and Breach; 1993.

10. Srivastava HM, Karlsson PW: Multiple Gaussian hypergeometric series. New York: Ellis Horwood, Chichester. John Wiley and Sons; 1985.

11. Wiman A: Uber den fundamental Satz in der Theorie der Funktionen E α ( x ). Acta Math 1905, 29: 191-201. 10.1007/BF02403202

12. Wright EM: The asymptotic expansion of generalized Bessel function. Proc London Math Soc 1934, 38(2):257-270.

13. Wright, EM: The asymptotic expansion of generalized hypergeometric functions. J London Math Soc 1935, 10: 286-293.

14. Wright, EM: The asymptotic expansion of generalized hypergeometric functions. J London Math Soc 1940a, 46(2):389-408.

15. Wright, EM: The asymptotic expansion of integral functions defined by Taylor series. Philos Trans Roy Soc London, Ser A 1940b, 238: 423-445. 10.1098/rsta.1940.0002

16. Wright, EM: The generalized Bessel function of order greater than one. Quart J Math Oxford Ser 1940c, 11: 36-48.

Acknowledgements

Author wish to thank refrees for valuable suggestions.

Author information

Correspondence to Shakeel Ahmed. 