Research Article Fractional Calculus of the Generalized Mittag-Leffler Type Functiondownloads.hindawi.com/archive/2014/907432.pdf · 2019-07-31 · Research Article Fractional Calculus

Research ArticleFractional Calculus of the Generalized Mittag-LefflerType Function

Dinesh Kumar1 and Sunil Kumar2

1 Department of Mathematics and Statistics, Jai Narain Vyas University, Jodhpur 342005, India2Department of Mathematics, National Institute of Technology Jamshedpur, Jharkhand 831014, India

Correspondence should be addressed to Dinesh Kumar; dinesh [email protected]

Received 22 March 2014; Accepted 6 May 2014; Published 18 August 2014

Academic Editor: Paolo Maria Mariano

Copyright © 2014 D. Kumar and S. Kumar. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We introduce and study a new function called 𝑅-function, which is an extension of the generalized Mittag-Leffler function. Wederive the relations that exist between the 𝑅-function and Saigo fractional calculus operators. Some results derived by Samko etal. (1993), Kilbas (2005), Kilbas and Saigo (1995), and Sharma and Jain (2009) are special cases of the main results derived in thispaper.

1. Introduction and Preliminaries

TheMittag-Leffler function has gained importance and pop-ularity during the last one and a half decades due to its directinvolvement in the problems of physics, biology, engineering,and applied sciences.Mittag-Leffler function naturally occursas the solution of fractional order differential equations andfractional order integral equations.

In 1903, the Swedish mathematician Mittag-Leffler [1, 2]introduced the function 𝐸

𝛼(𝑧), defined by

𝐸𝛼(𝑧) =

∞

∑

𝑛=0

𝑧𝑛

Γ (𝛼𝑛 + 1), (𝛼 ∈ 𝐶,Re (𝛼) > 0) . (1)

The Mittag-Leffler function 𝐸𝛼(𝑧) was studied by Wiman [3]

who defined the function 𝐸𝛼,𝛽(𝑧) as follows:

𝐸𝛼,𝛽(𝑧) =

∞

∑

𝑛=0

𝑧𝑛

Γ (𝛼𝑛 + 𝛽),

(𝛼, 𝛽 ∈ 𝐶,Re (𝛼) > 0,Re (𝛽) > 0) .

(2)

The function 𝐸𝛼,𝛽(𝑧) is now known as Wiman function,

which was later studied by Agarwal [4] and others.

The generalization of (2) was introduced by Prabhakar [5]in terms of the series representation

𝐸𝛾

𝛼,𝛽(𝑧) =

∞

∑

𝑛=0

(𝛾)𝑛𝑧𝑛

Γ (𝛼𝑛 + 𝛽) 𝑛!,

(𝛼, 𝛽, 𝛾 ∈ 𝐶,Re (𝛼) > 0,Re (𝛽) > 0) ,

(3)

where (𝛾)𝑛is Pochhammer’s symbol, defined by

(𝛾)𝑛=Γ (𝛾 + 𝑛)

Γ (𝛾)

= {1, (𝑛 = 0, 𝛾 = 0)

𝛾 (𝛾 + 1) ⋅ ⋅ ⋅ (𝛾 + 𝑛 − 1) , (𝑛 ∈ 𝑁, 𝛾 ∈ 𝐶) .

(4)

Shukla and Prajapati [6] defined and investigated the func-tion 𝐸𝛾,𝑞

𝛼,𝛽(𝑧) as

𝐸𝛾,𝑞

𝛼,𝛽(𝑧) =

∞

∑

𝑛=0

(𝛾)𝑞𝑛𝑧𝑛

Γ (𝛼𝑛 + 𝛽) 𝑛!,

(𝛼, 𝛽, 𝛾 ∈ 𝐶,Re (𝛼) > 0,Re (𝛽) > 0,Re (𝛾) > 0) ,

(5)

Hindawi Publishing CorporationInternational Scholarly Research NoticesVolume 2014, Article ID 907432, 5 pageshttp://dx.doi.org/10.1155/2014/907432

2 International Scholarly Research Notices

where 𝑞 ∈ (0, 1) ∪ 𝑁 and (𝛾)𝑞𝑛= Γ(𝛾 + 𝑞𝑛)/Γ(𝛾) denotes the

generalized Pochhammer symbol which in particular reducesto

𝑞𝑞𝑛

𝑞

∏

𝑟=1

(𝛾 + 𝑟 − 1

𝑞)

𝑛

, 𝑞 ∈ 𝑁. (6)

Srivastava and Tomovski [7] introduced and investigated afurther generalization of (3), which is defined in the followingway:

𝐸𝛾,𝑘

𝛼,𝛽(𝑧) =

∞

∑

𝑛=0

(𝛾)𝑘𝑛𝑧𝑛

Γ (𝛼𝑛 + 𝛽) 𝑛!,

(𝑧, 𝛽, 𝛾 ∈ 𝐶;Re (𝛼) > max {0,Re (𝑘) − 1} ;Re (𝑘) > 0) ,

(7)

which, in the special case when 𝑘 = 𝑞 (𝑞 ∈ (0, 1) ∪ 𝑁) andmin{Re(𝛽),Re(𝛾)} > 0, is given by (5).

It is an entire function of order 𝜌 = [Re(𝛼)]−1. Some spe-cial cases of (3) are

𝐸𝛼(𝑧) = 𝐸

1

𝛼,1(𝑧) , 𝐸

𝛼,𝛽(𝑧) = 𝐸

1

𝛼,𝛽(𝑧) ,

𝜙 (𝛽, 𝛾; 𝑧) =1𝐹1(𝛽, 𝛾; 𝑧) = Γ (𝛾) 𝐸

𝛽

1,𝛾(𝑧) ;

(8)

here1𝐹1denotes an hypergeometric function; see also

2𝐹1in

(10).

Remark 1. A detailed account ofMittag-Leffler functions andtheir applications can be found in themonograph byHauboldet al. [8].

An interesting generalization of (2) is recently introducedby Kilbas and Saigo [9] in terms of a special entire functionas given below:

𝐸𝛼,𝑚,𝑟

(𝑧) =

∞

∑

𝑛=0

𝑐𝑛𝑧𝑛

, (9)

where 𝑐𝑛= ∏𝑛−1

𝑖=0(Γ[𝛼(𝑖𝑚 + 𝑟) + 1]/Γ[𝛼(𝑖𝑚 + 𝑟 + 1) + 1]) and

an empty product is to be interpreted as unity.

1.1. Fractional Integrals and Derivatives. An interesting anduseful generalization of both the Riemann-Liouville andErdelyi-Kober fractional integration operators is introducedby Saigo [10] in terms of Gauss’s hypergeometric function asfollows:

𝐼𝛼,𝛽,𝜂

0+𝑓 (𝑧)

=𝑧−𝛼−𝛽

Γ (𝛼)∫

𝑧

0

(𝑧 − 𝑡)𝛼−1

2𝐹1(𝛼 + 𝛽, −𝜂; 𝛼; 1 −

𝑡

𝑧)

× 𝑓 (𝑡) 𝑑𝑡,

(10)

where 𝛼, 𝛽, 𝜂 ∈ 𝐶, Re(𝛼) > 0, and 𝑧 ∈ 𝑅+, and the generalized

fractional derivative of a function

𝐷𝛼,𝛽,𝜂

0+𝑓 (𝑧)

=𝑧𝛼+𝛽

Γ (𝑛 − 𝛼)(𝑑

𝑑𝑥)

𝑛

× ∫

𝑧

0

(𝑧 − 𝑡)𝑛−𝛼−1

×2𝐹1(−𝛼 − 𝛽, 𝑛 − 𝛼 − 𝜂; 𝑛 − 𝛼; 1 −

𝑡

𝑧)𝑓 (𝑡) 𝑑𝑡,

(11)

where 𝑛 = [Re(𝛼)] + 1.

2. The 𝑅-Function

The 𝑅-function is introduced by the authors as follows:

𝑘

𝑝𝑅𝛼,𝛽;𝛾

𝑞

(𝑧) =𝑘


𝑞

(𝑎1, ..., 𝑎𝑝; 𝑏1, ..., 𝑏𝑞; 𝑧)

=

∞

∑

𝑛=0

∏𝑝

𝑗=1(𝑎𝑗)𝑛

∏𝑞

𝑗=1(𝑏𝑗)𝑛


𝑛!Γ (𝛼𝑛 + 𝛽),

(12)

where 𝛼, 𝛽, 𝛾 ∈ 𝐶, Re(𝛼) > max{0,Re(𝑘) − 1}; Re(𝑘) > 0;(𝑎𝑗)𝑛

, and (𝑏𝑗)𝑛

are the Pochhammer symbols. The series (12)is defined when none of the parameters 𝑏

𝑗’s, 𝑗 = 1, 𝑞, is a

negative integer or zero. If any parameter 𝑎𝑗is a negative

integer or zero, then the series (12) terminates to a polynomialin, and the series is convergent for all 𝑧 if 𝑝 < 𝑞+1. It can alsoconverge in some cases if we have 𝑝 = 𝑞 + 1 and |𝑧| = 1.Let 𝛾 = ∑𝑝

𝑗=1𝑎𝑗− ∑𝑞

𝑗=1𝑏𝑗; it can be shown that if Re(𝛾) > 0

and 𝑝 = 𝑞 + 1 the series is absolutely convergent for |𝑧| = 1,in order convergent for 𝑧 = −1 when 0 ≤ Re(𝛾) < 1 anddivergent for |𝑧| = 1 when 1 ≤ Re(𝛾).

2.1. Special Cases of the 𝑅-Function. (i) If we set 𝑎𝑗= 𝑏𝑗= 1,

we have

𝑘

0𝑅𝛼,𝛽;𝛾

0(𝑧) =

∞

∑

𝑛=0


𝑛!Γ (𝛼𝑛 + 𝛽)= 𝐸𝛾,𝑘

𝛼,𝛽(𝑧) , (13)

where 𝐸𝛾,𝑘𝛼,𝛽(𝑧) is the generalized Mittag-Leffler function

introduced by Srivastava and Tomovski [7]; compare (5).(ii) In the special case of (13), when 𝑘 = 𝑞 (𝑞 ∈ (0, 1)∪𝑁)

and min{Re(𝛽),Re(𝛾)} > 0, we have the following:

𝑞

0𝑅𝛼,𝛽;𝛾

0(𝑧) =

∞

∑

𝑛=0

(𝛾)𝑞𝑛𝑧𝑛

𝑛!Γ (𝛼𝑛 + 𝛽)= 𝐸𝛾,𝑞

𝛼,𝛽(𝑧) , (14)

where 𝐸𝛾,𝑞𝛼,𝛽(𝑧) was considered earlier by Shukla and Prajapati

[6].

International Scholarly Research Notices 3

(iii) If we set 𝑎𝑗= 𝑏𝑗= 𝑘 = 1 in (12), we have

1

0𝑅𝛼,𝛽;𝛾

0(𝑧) =

∞

∑

𝑛=0

(𝛾)𝑛𝑧𝑛

𝑛!Γ (𝛼𝑛 + 𝛽)= 𝐸𝛾

𝛼,𝛽(𝑧) , (15)

where 𝐸𝛾𝛼,𝛽(𝑧) is generalization of the Mittag-Leffler function

introduced by Prabhakar [5] and studied byHaubold et al. [8]and others; compare (3).

(iv) If we put 𝛾 = 1 in (15), we have

1

0𝑅𝛼,𝛽;1

0(𝑧) =

∞

∑

𝑛=0

𝑧𝑛

Γ (𝛼𝑛 + 𝛽)= 𝐸1

𝛼,𝛽(𝑧) = 𝐸

𝛼,𝛽(𝑧) , (16)

where 𝐸𝛼,𝛽(𝑧) is the generalized Mittag-Leffler function [3]

(also known as Wiman function), which was later studied byAgarwal [4] and others; compare (2).

(v) If we take 𝛽 = 𝛾 = 1 in (15), we have

1

0𝑅𝛼,1;1

0(𝑧) =

∞

∑

𝑛=0

𝑧𝑛

Γ (𝛼𝑛 + 1)= 𝐸1

𝛼,1(𝑧) = 𝐸

𝛼(𝑧) , (17)

where 𝐸𝛼(𝑧) is the Mittag-Leffler function [1, 2]; compare (1).

(vi) If we take 𝛼 = 𝛽 = 𝛾 = 1 in (15), we obtain

1

0𝑅1,1;1

0(𝑧) =

∞

∑

𝑛=0

𝑧𝑛

Γ (𝑛 + 1)= 𝐸1

1,1(𝑧) = 𝐸

1(𝑧) = 𝑒

𝑥

, (18)

where 𝑒𝑥 is the exponential function [11].(vii) If we set 𝛾 = 𝑘 = 1 in (12), then the 𝑅-function can

be represented in the Wright generalized hypergeometricfunction [12]

𝑝

𝜓𝑞

(𝑧) and the 𝐻-function [13, 14] as givenbelow:

1

𝑝𝑅𝛼,𝛽;1

𝑞

(𝑧)

=1

𝑝𝑅𝛼,𝛽;1

𝑞

(𝑎1, . . . , 𝑎

𝑝; 𝑏1, . . . , 𝑏

𝑞; 𝑧)

= Ω𝑝+1

𝜓𝑞+1

[𝑧|(𝑎1,1),...,(𝑎

𝑝,1),(1,1)

(𝑏1,1),...,(𝑏

𝑞,1),(𝛽,𝛼)

]

= Ω𝐻1,𝑝+1

𝑝+1,𝑞+2[−𝑧|(1−𝑎𝑗,1)1,𝑝,(0,1)

(0,1),(1−𝑏𝑗,1)1,𝑞,(1−𝛽,𝛼)

] ,

where∏𝑞

𝑗=1Γ(𝑏𝑗)𝑟

∏𝑝

𝑗=1Γ(𝑎𝑗)𝑟

,

(19)

where𝐻-function is as defined in the monograph by Mathaiand Saxena [14].

(viii) If we set 𝑝 = 𝑞 = 0 and 𝛾 = 𝑘 = 1 in (12), thenwe obtain another special case of 𝑅-function in terms of theWright generalized hypergeometric function as given below:

1

0𝑅𝛼,𝛽;1

0(𝑧) =

1

0𝑅𝛼,𝛽;1

0(−; 1; 𝑧) =

∞

∑

𝑛=0

Γ (𝑛 + 1) 𝑧𝑛

Γ (𝛼𝑛 + 𝛽) 𝑛!

=(1)𝑛𝑧𝑛

Γ (𝛼𝑛 + 𝛽) 𝑛!=1

𝜓1[𝑧|(1,1)

(𝛽,𝛼)] .

(20)

(ix) If we set 𝛼 = 𝛽 = 𝛾 = 𝑘 = 1 in (12), then the𝑅-function reduces to the generalized hypergeometric func-tion𝑝𝐹𝑞(see for detail [11, 15, 16]) as given below:

1

𝑝𝑅1,1;1

𝑞

(𝑎1, . . . , 𝑎

𝑝; 𝑏1, . . . , 𝑏

𝑞; 𝑧)

=

∞

∑

𝑛=0

∏𝑝


∏𝑞


𝑧𝑛

𝑛!=𝑝𝐹𝑞

((𝑎𝑗)1,𝑝

; (𝑏𝑗)1,𝑞

; 𝑧) .

(21)

3. Relations with Generalized FractionalCalculus Operators

In this section we derive two theorems relating to generalizedfractional integrals and derivative of the 𝑅-function.

Theorem 2. Let 𝜗, 𝜂, 𝛿, 𝛼, 𝛽, 𝛾 ∈ 𝐶, Re(𝜗) > 0, Re(𝛼) > 0, and(𝑎)𝑛= Γ(𝑎 + 𝑛)/Γ(𝑎); then there holds the relation

𝐼𝜗,𝜂,𝛿

0+(𝑘


𝑞

(𝑥))

=𝑥−𝜂

Γ (1 + 𝛿 − 𝜂)

Γ (1 + 𝜗 + 𝛿) Γ (1 − 𝜂)

×𝑘

𝑝+2𝑅𝛼,𝛽;𝛾

𝑞+2

(𝑎1, . . . , 𝑎

𝑝, 1, (1 + 𝛿 − 𝜂) ; 𝑏

1, . . . , 𝑏

𝑞,

(1 + 𝜗 + 𝛿) , (1 − 𝜂) ; 𝑥) .

(22)

Proof. Following the definition of Saigo fractional integral[17] as given in (10), we have the following relation:

𝐼𝜗,𝜂,𝛿

0+(𝑘


𝑞(𝑥))

=𝑥−𝜗−𝜂

Γ (𝜗)∫

𝑥

0

(𝑥 − 𝑡)𝜗−1

2𝐹1(𝜗 + 𝜂, −𝛿; 𝜗; 1 −

𝑡

𝑥)

×𝑘


𝑞

(𝑡) 𝑑𝑡;

(23)

by virtue of (12), we obtain

𝐼𝜗,𝜂,𝛿

0+(𝑘


𝑞(𝑥))

=𝑥−𝜗−𝜂

Γ (𝜗)∫

𝑥

0

(𝑥 − 𝑡)𝜗−1

2𝐹1(𝜗 + 𝜂, −𝛿; 𝜗; 1 −

𝑡

𝑥)

×

∞

∑

𝑛=0

∏𝑝


∏𝑞


(𝛾)𝑘𝑛𝑡𝑛

𝑛!Γ (𝛼𝑛 + 𝛽)𝑑𝑡.

(24)

4 International Scholarly Research Notices

Interchanging the order of integration and evaluating theinner integral with the help of Beta function, we arrive at

𝐼𝜗,𝜂,𝛿

0+(𝑘


𝑞

(𝑥))

=𝑥−𝜂

Γ (1 + 𝛿 − 𝜂)

Γ (1 + 𝜗 + 𝛿) Γ (1 − 𝜂)

×

∞

∑

𝑛=0

∏𝑝


(1)𝑛(𝛿 − 𝜂 + 1)

𝑛

∏𝑞


(𝜗 + 𝛿 + 1)𝑛(1 − 𝜂)

𝑛

(𝛾)𝑘𝑛𝑥𝑛

𝑛!Γ (𝛼𝑛 + 𝛽)

=𝑥−𝜂

Γ (1 + 𝛿 − 𝜂)

Γ (1 + 𝜗 + 𝛿) Γ (1 − 𝜂)

×𝑘


𝑞+2

(𝑎1, . . . , 𝑎

𝑝, 1, (𝛿 − 𝜂 + 1) ; 𝑏

1, . . . , 𝑏

𝑞,

(𝜗 + 𝛿 + 1) , (1 − 𝜂) ; 𝑥) .

(25)

The interchange of the order of summation is permissibleunder the conditions stated along with the theorem. Thisshows that a Saigo fractional integral of the 𝑅-function isagain the 𝑅-function with increased order (𝑝 + 2, 𝑞 + 2).

This completes the proof of Theorem 2.

If we put 𝜂 = −𝜗, then we obtain following Corollary con-cerning Riemann-Liouville fractional integral operator [16].

Corollary 3. Let 𝜗, 𝛼, 𝛽, 𝛾 ∈ 𝐶, Re(𝜗) > 0, Re(𝛼) > 0, and(𝑎)𝑛= Γ(𝑎 + 𝑛)/Γ(𝑎), then there holds the relation

𝐼𝜗

0+(𝑘


𝑞

(𝑥))

=𝑥𝜗

Γ (1 + 𝜗)

𝑘


𝑞+1

(𝑎1, . . . , 𝑎

𝑝, 1; 𝑏1, . . . , 𝑏

𝑞, (1 + 𝜗) ; 𝑥) .

(26)

Theorem 4. Let 𝜗, 𝜂, 𝛿, 𝛼, 𝛽, 𝛾 ∈ 𝐶, Re(𝜗) > 0, Re(𝛼) > 0, and(𝑎)𝑛= Γ(𝑎 + 𝑛)/Γ(𝑎), then there holds the relation

𝐷𝜗,𝜂,𝛿

0+(𝑘


𝑞

(𝑥))

=𝑥𝜂

Γ (1 + 𝜗 + 𝜂 + 𝛿)

Γ (1 + 𝛿) Γ (1 + 𝜂)

×𝑘


𝑞+2

(𝑎1, . . . , 𝑎

𝑝, 1, (1 + 𝜗 + 𝜂 + 𝛿) ;

𝑏1, . . . , 𝑏

𝑞, (1 + 𝛿) , (1 + 𝜂) ; 𝑥) .

(27)

Proof. Following the definition of Saigo fractional derivativeas given in (11), we have the following relation:

𝐷𝜗,𝜂,𝛿

0+(𝑘


𝑞

(𝑥))

=𝑥𝜗+𝜂

Γ (𝑟 − 𝜗)(𝑑

𝑑𝑥)

𝑟

× ∫

𝑥

0

(𝑥 − 𝑡)𝑟−𝜗−1

2𝐹1(−𝜗 − 𝜂, 𝑟 − 𝜗 − 𝛿; 𝑟 − 𝜗; 1 −

𝑡

𝑥)

×𝑘

𝑝𝑅𝛼,𝛽,𝛾

𝑞

(𝑡) 𝑑𝑡,

(28)

where 𝑟 = [Re(𝜗)] + 1.By virtue of (12), we obtain

𝐷𝜗,𝜂,𝛿

0+(𝑘


𝑞(𝑥))

=𝑥𝜗+𝜂

Γ (𝑟 − 𝜗)(𝑑

𝑑𝑥)

𝑟

× ∫

𝑥

0

(𝑥 − 𝑡)𝑟−𝜗−1

2𝐹1(−𝜗 − 𝜂, 𝑟 − 𝜗 − 𝛿; 𝑟 − 𝜗; 1 −

𝑡

𝑥)

×

∞

∑

𝑛=0

∏𝑝


∏𝑞


(𝛾)𝑘𝑛𝑡𝑛

𝑛!Γ (𝛼𝑛 + 𝛽)𝑑𝑡.

(29)

Interchanging the order of integration and evaluating theinner integral with the help of Beta function, we arrive at

𝐷𝜗,𝜂,𝛿

0+(𝑘


𝑞

(𝑥))

=𝑥𝑟+𝜂

Γ (1 + 𝜗 + 𝜂 + 𝛿)

Γ (1 + 𝛿) Γ (1 + 𝑟 + 𝜂)

∞

∑

𝑛=0

∏𝑝


(1)𝑛(1 + 𝜗 + 𝜂 + 𝛿)

𝑛

∏𝑞


(1 + 𝛿)𝑛(1 + 𝑟 + 𝜂)

𝑛

×(𝛾)𝑘𝑛𝑥𝑛

𝑛!Γ (𝛼𝑛 + 𝛽)

=𝑥𝑟+𝜂

Γ (1 + 𝜗 + 𝜂 + 𝛿)

Γ (1 + 𝛿) Γ (1 + 𝑟 + 𝜂)

×𝑘


𝑞+2

(𝑎1, . . . , 𝑎

𝑝, 1, (1 + 𝜗 + 𝜂 + 𝛿) ; 𝑏

1, . . . , 𝑏

𝑞,

(1 + 𝛿) , (1 + 𝑟 + 𝜂) ; 𝑥) .

(30)

Here 𝑟 = [Re(𝜗)] + 1, and by using (𝑑𝑟/𝑑𝑥𝑟)𝑥𝑚 = (Γ(𝑚 +1)/Γ(𝑚 − 𝑟 + 1))𝑥

𝑚−𝑟, where 𝑚 ≥ 𝑟 in the above expression,we obtain the right-hand side of (27).This shows that a Saigofractional derivative of the𝑅-function is again the𝑅-functionwith increased order (𝑝 + 2, 𝑞 + 2).

This completes the proof of Theorem 4.

If we put 𝜂 = −𝜗, then we obtain following Corollaryconcerning Riemann-Liouville fractional derivative operator[16].

International Scholarly Research Notices 5

Corollary 5. Let 𝜗, 𝛼, 𝛽, 𝛾 ∈ 𝐶, Re(𝜗) > 0, Re(𝛼) > 0, and(𝑎)𝑛= Γ(𝑎 + 𝑛)/Γ(𝑎); then there holds the relation

𝐷𝜗

0+(𝑘


𝑞(𝑥))

=𝑥−𝜗

Γ (1 − 𝜗)

𝑘


𝑞+1

(𝑎1, . . . , 𝑎

𝑝, 1; 𝑏1, . . . , 𝑏

𝑞, (1 − 𝜗) ; 𝑥) .

(31)

Remark 6. A number of known and new results can beobtained as special cases of Theorems 2 and 4, but we do notmention them here on account of lack of space.

4. Conclusion

In this paper we derive a new generalization of Mittag-Lefflerfunction and obtain the relations between the 𝑅-functionand Saigo fractional calculus operators. The results are alsoextension of work done by Sharma [18]. The provided resultsare new and have uniqueness identity in the literature. Anumber of known and new results are special cases of ourmain findings.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The authors wish to thank the referees for their usefulsuggestions for the improvement of the paper.The authors arethankful to Professor R. K. Saxena for giving useful sugges-tions, which led to the present form of the paper.

References

[1] G. M. Mittag-Leffler, “Sur la nouvelle fonction E𝛼(x),” Comptes

Rendus de l’Academie des Sciences Paris, vol. 137, pp. 554–558,1903.

[2] G. M. Mittag-Leffler, “Sur la representation analytique d’unebranche uniforme d’une function monogene,” Acta Mathemat-ica, vol. 29, pp. 101–181, 1905.

[3] A.Wiman, “Uber den Fundamentalsatz in der Teorie der Funk-tionen E

𝛼(x),”ActaMathematica, vol. 29, no. 1, pp. 191–201, 1905.

[4] R. P. Agarwal, “A propos d’une note de M. Pierre Humbert,”Comptes Rendus de l’Academie des Sciences, Paris, vol. 236, pp.2031–2032, 1953.

[5] T. R. Prabhakar, “A singular integral equationwith a generalizedMittag-Leffler function in the Kernel,” Yokohama Mathemati-cal Journal, vol. 19, pp. 7–15, 1971.

[6] A. K. Shukla and J. C. Prajapati, “On a generalization of Mittag-Leffler function and its properties,” Journal of MathematicalAnalysis and Applications, vol. 336, no. 2, pp. 797–811, 2007.

[7] H. M. Srivastava and Z. Tomovski, “Fractional calculus withan integral operator containing a generalized Mittag-Lefflerfunction in the kernel,” Applied Mathematics and Computation,vol. 211, no. 1, pp. 198–210, 2009.

[8] H. J. Haubold, A. M. Mathai, and R. K. Saxena, “Mittag-lefflerfunctions and their applications,” Journal of Applied Mathemat-ics, vol. 2011, Article ID 298628, 51 pages, 2011.

[9] A. A. Kilbas and M. Saigo, “Fractional integrals and deriva-tives of Mittag-Leffler type function,” Doklady Akademii NaukBelarusi, vol. 39, no. 4, pp. 22–26, 1995.

[10] M. Saigo, “A remark on integral operators involving the Gausshypergeometric functions,” Mathematical Reports, College ofGeneral Education, Kyushu University, vol. 11, pp. 135–143, 1978.

[11] E. D. Rainville, Special Functions, Chelsea Publishing Company,Bronx, NY, USA, 1960.

[12] E. M.Wright, “The asymptotic expansion of generalized hyper-geometric function,” Journal of the London Mathematical Soci-ety, vol. 10, pp. 286–293, 1935.

[13] A. A. Kilbas, “Fractional calculus of the generalized Wrightfunction,” Fractional Calculus and Applied Analysis, vol. 8, no.2, pp. 113–126, 2005.

[14] A. M. Mathai and R. K. Saxena, The H-function with Applica-tions in Statistics and Other Disciplines, John Wiley and Sons,New York, NY, USA, 1978.

[15] C. F. Lorenzo and T. T. Hartley, “Generalized function for thefractional calculus,” Tech. Rep. NASA/TP-1999-209424, 1999.

[16] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Inte-grals and Derivatives, Theory and Applications, Gordon andBreach, Yverdon et alibi, 1993.

[17] H. M. Srivastava and R. K. Saxena, “Operators of fractionalintegration and their applications,” Applied Mathematics andComputation, vol. 118, no. 1, pp. 1–52, 2001.

[18] K. Sharma, “Application of fractional calculus operators torelated areas,” General Mathematics Notes, vol. 7, no. 1, pp. 33–40, 2011.

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