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The extended Mittag-Leffler function and its properties

Abstract

In this paper, we present the extended Mittag-Leffler functions by using the extended Beta functions (Chaudhry et al. in Appl. Math. Comput. 159:589-602, 2004) and obtain some integral representations of them. The Mellin transform of these functions is given in terms of generalized Wright hypergeometric functions. Furthermore, we show that the extended fractional derivative (Özarslan and Özergin in Math. Comput. Model. 52:1825-1833, 2010) of the usual Mittag-Leffler function gives the extended Mittag-Leffler function. Finally, we present some relationships between these functions and the Laguerre polynomials and Whittaker functions.

1 Introduction

Fractional differential equations have been an active research area during the past few decades and they occur in many applications of physics and engineering. The Mittag-Leffler function appears as the solution of fractional order differential equations and fractional order integral equations. Some applications of the Mittag-Leffler function are as follows: studies of the kinetic equation, the telegraph equation [1], random walks, Levy flights, superdiffuse transport, and complex systems. Besides this, the Mittag-Leffler function appears in the solution of certain boundary value problems involving fractional integro-differential equations of Volterra type [2]. It has applications in applied problems, such as fluid flow, rheology, diffusive transport akin to diffusion, electric networks, probability, and statistical distribution theory. Various properties of the Mittag-Leffler functions were presented and surveyed in [3]. Furthermore, a different variant of the Mittag-Leffler function has been investigated in [4].

Let us start with giving the historical background of the Mittag-Leffler functions. The function E α (z),

E α (z)= k = 0 z k Γ ( α k + 1 ) ,
(1)

was defined and studied by Mittag-Leffler in the year 1903 in [57]. It is a direct generalization of the exponential series, since, for α=1, we have the exponential function. The function defined by

E α , β (z)= k = 0 z k Γ ( α k + β )
(2)

gives a generalization of equation (1). This generalization was studied by Wiman in 1905 [8, 9], Agarwal in 1953, and Humbert and Agarwal [10, 11] in 1953. Afterward, Prabhakar [12] introduced the generalized Mittag-Leffler function by

E β , γ δ (z):= n = 0 ( δ ) n Γ ( β n + γ ) z n n ! ,
(3)

where β,γ,δC with (β)>0. For δ=1, it reduces to the Mittag-Leffler function given in equation (2). Some of the properties of the generalized Mittag-Leffler function such as the Mellin transform, the inverse Mellin transform, and differentiation were given in [13]. On the other hand, monotony of the Mittag-Leffler function was given in [14].

In this paper, we extend the Mittag-Leffler function E α , β γ (z) in the following way. Since

E α , β γ (z)= k = 0 ( γ ) k Γ ( α k + β ) ( c ) k ( c ) k z k k ! ,

using the fact that

( γ ) k ( c ) k = B ( γ + k , c γ ) B ( γ , c γ ) ,

we extend the Mittag-Leffler function as follows:

E α , β ( γ ; c ) ( z ; p ) : = k = 0 B p ( γ + k , c γ ) B ( γ , c γ ) ( c ) k Γ ( α k + β ) z k k ! ( p 0 ; Re ( c ) > Re ( γ ) > 0 ) ,
(4)

where for B p (x,y) we have

B p (x,y)= 0 1 t x 1 ( 1 t ) y 1 e p t ( 1 t ) dt ( Re ( p ) > 0 , Re ( x ) > 0 , Re ( y ) > 0 ) ,
(5)

the extended Euler’s Beta function defined in [15] (see also [16]).

The organization of the paper is as follows: In Section 2, we give an integral representations of the extended Mittag-Leffler function in terms of Prabhakar’s Mittag-Leffler function and in terms of known elementary functions. The Mellin transform of the extended Mittag-Leffler function is obtained by means of the generalized Wright hypergeometric function [17]. In Section 3, we obtain fractional derivative representations of the extended Mittag-Leffler function and give some derivative formulas. In Section 4, we obtain the relationship between the extended Mittag-Leffler function and simple Laguerre polynomials and Whittaker’s functions.

2 Some properties of the extended Mittag-Leffler function

We begin with the following theorem, which gives the integral representation of the extended Mittag-Leffler function.

Theorem 1 (Integral representation)

For the extended Mittag-Leffler function, we have

E α , β ( γ ; c ) (z;p)= 1 B ( γ , c γ ) 0 1 t γ 1 ( 1 t ) c γ 1 e p t ( 1 t ) E α , β ( c ) (tz)dt,
(6)

where p0, Re(c)>Re(γ)>0, Re(α)>0, Re(β)>0.

Proof Using equation (5) in equation (4), we get

E α , β ( γ ; c ) (z;p)= k = 0 { 0 1 t γ + k 1 ( 1 t ) c γ 1 e p t ( 1 t ) d t } ( c ) k B ( γ , c γ ) z k Γ ( α k + β ) k ! .
(7)

Interchanging the order of summation and integration in equation (7), which is guaranteed under the assumptions given in the statement of the theorem, we get

E α , β ( γ ; c ) ( z ; p ) = 0 1 t γ 1 ( 1 t ) c γ 1 e p t ( 1 t ) k = 0 ( c ) k B ( γ , c γ ) ( t z ) k Γ ( α k + β ) k ! d t .
(8)

Using equation (3) in equation (8), we get the desired result. □

Corollary 2 Note that, taking t= u 1 + u in Theorem  1, we get

E α , β ( γ ; c ) ( z ; p ) = 1 B ( γ , c γ ) 0 u γ 1 ( u + 1 ) c e p ( 1 + u ) 2 u E α , β ( c ) ( u z 1 + u ) d u .
(9)

Corollary 3 Taking t= sin 2 θ in the Theorem  1, we get the following integral representation:

E α , β ( γ ; c ) ( z ; p ) = 1 B ( γ , c γ ) [ 2 0 π 2 sin 2 γ 1 θ cos 2 c 2 γ 1 θ e p sin 2 θ cos 2 θ ] × E α , β ( c ) ( z sin 2 θ ) d θ .
(10)

Now, using the definition of Prabhakar’s Mittag-Leffler’s function, Bayram and Kurulay obtained the recurrence formula [13]

E α , β ( c ) (tz)=β E α , β + 1 ( c ) (tz)+αz d d z E α , β + 1 ( c ) (tz).

Inserting the above recurrence relation into equation (6), we get the following recurrence relation for the extended Mittag-Leffler’s function.

Corollary 4 (Recurrence relation)

For the extended Mittag-Leffler function, we get

E α , β ( γ ; c ) (z;p)=β E α , β + 1 ( γ ; c ) (z;p)+αz d d z E α , β + 1 ( γ ; c ) (z;p),

where p0, Re(c)>Re(γ)>0, Re(α)>0, Re(β)>0.

In the next theorem, we give the Mellin transform of the extended Mittag-Leffler function in terms of the Wright generalized hypergeometric function. Note that the Wright generalized hypergeometric function is defined by [17]

Ψ q p ( z ) = Ψ q p [ ( a 1 , A 1 ) , ( a 2 , A 2 ) , , ( a p , A p ) ( b 1 , B 1 ) , ( b 2 , B 2 ) , , ( b p , B q ) , z ] = k = 0 j = 1 p Γ ( a j + A j k ) j = 1 q Γ ( b j + B j k ) z k k ! ,
(11)

where the coefficients A i (i=1,,p) and B j (j=1,,q) are positive real numbers such that

1+ j = 1 q B j i = 1 p A i 0.

Theorem 5 (Mellin transform)

The Mellin transform of the extended Mittag-Leffler function is given by

M { E α , β ( γ ; c ) ( z ; p ) ; s } = Γ ( s ) Γ ( c + s γ ) Γ ( γ ) Γ ( c γ ) 2 Ψ 2 [ ( c , 1 ) , ( β , γ ) , ( γ + s , 1 ) ( c + 2 s , 1 ) , z ] ( p 0 , Re ( c ) > Re ( γ ) > 0 , Re ( α ) > 0 , Re ( s ) > 0 , Re ( β ) > 0 ) ,
(12)

where Ψ 2 2 is the Wright generalized hypergeometric function.

Proof Taking the Mellin transform of the extended Mittag-Leffler function, we have

M { E α , β ( γ ; c ) ( z ; p ) ; s } = 0 p s 1 E α , β ( γ ; c ) (z;p)dp.
(13)

Using equation (6) in equation (13), we get

M { E α , β ( γ ; c ) ( z ; p ) ; s } = 1 B ( γ , c γ ) 0 p s 1 [ 0 1 t γ 1 ( 1 t ) c γ 1 e p t ( 1 t ) ] E α , β ( c ) ( t z ) d t d p .
(14)

Interchanging the order of integrals in equation (14), which is valid because of the conditions in the statement of the Theorem 5, we get

M { E α , β ( γ ; c ) ( z ; p ) ; s } = 1 B ( γ , c γ ) 0 1 [ t γ 1 ( 1 t ) c γ 1 E α , β ( c ) ( t z ) ] 0 p s 1 e p t ( 1 t ) d p d t .
(15)

Now taking u= p t ( 1 t ) in equation (15) and using the fact that Γ(s)= 0 u s 1 e u du, we get

M { E α , β ( γ ; c ) ( z ; p ) ; s } = Γ ( s ) B ( γ , c γ ) 0 1 t γ + s 1 ( 1 t ) c + s γ 1 E α , β ( c ) ( t z ) d t .
(16)

Using the definition of Prabhakar’s generalized Mittag-Leffler function E α , β ( c ) (tz) in equation (16), we get

M { E α , β ( γ ; c ) ( z ; p ) ; s } = Γ ( s ) B ( γ , c γ ) 0 1 t γ + s 1 ( 1 t ) c + s γ 1 k = 0 ( c ) k ( t z ) k Γ ( α k + β ) k ! d t .
(17)

Interchanging the order of summation and integration, which is valid for Re(c)>Re(γ)>0, Re(s)>0, Re(cγ+s)>0, Re(α)>0, Re(β)>0, we get

M { E α , β ( γ ; c ) ( z ; p ) ; s } = Γ ( s ) B ( γ , c γ ) k = 0 ( c ) k z k Γ ( α k + β ) k ! 0 1 t γ + k + s 1 ( 1 t ) c + s γ 1 d t .
(18)

Using the Beta function in equation (18), we have

M { E α , β ( γ ; c ) ( z ; p ) ; s } = Γ ( s ) Γ ( c + s γ ) B ( γ , c γ ) k = 0 ( c ) k z k Γ ( α k + β ) k ! Γ ( γ + k + s ) Γ ( γ + s ) Γ ( γ + s ) Γ ( c + k + 2 s ) .
(19)

Considering that ( c ) k = Γ ( c + k ) Γ ( c ) , B(γ,cγ)= Γ ( γ ) Γ ( c γ ) Γ ( c ) , and inserting equation (11) into equation (19), we get the result

M { E α , β ( γ ; c ) ( z ; p ) ; s } = Γ ( s ) Γ ( c + s γ ) B ( γ , c γ ) Γ ( c ) k = 0 z k Γ ( α k + β ) k ! Γ ( γ + k + s ) Γ ( c + k ) Γ ( c + k + 2 s ) = Γ ( s ) Γ ( c + s γ ) Γ ( γ ) Γ ( c γ ) 2 Ψ 2 [ ( c , 1 ) , ( β , α ) , ( γ + s , 1 ) ( c + 2 s , 1 ) , z ] .

 □

Corollary 6 Taking s=1 in Theorem  5, we get

0 E α , β ( γ ; c ) (z;p)dp= Γ ( c + 1 γ ) Γ ( γ ) Γ ( c γ ) 2 Ψ 2 [ ( c , 1 ) , ( β , α ) , ( γ + 1 , 1 ) ( c + 2 , 1 ) , z ] .

Corollary 7 Taking the inverse Mellin transform on both sides of equation (12), we get the elegant complex integral representation

E α , β ( γ ; c ) (z;p)= 1 2 π i Γ ( γ ) Γ ( c γ ) ν i ν + i Γ(s)Γ ( c + s γ ) 2 Ψ 2 [ ( c , 1 ) , ( β , α ) , ( γ + s , 1 ) ( c + 2 s , 1 ) , z ] p s ds,

where ν>0.

3 Derivative properties of the extended Mittag-Leffler function

The classical Riemann-Liouville fractional derivative of order μ is usually defined by

D z μ { f ( z ) } = 1 Γ ( μ ) 0 z f(t) ( z t ) μ 1 dt,Re(μ)<0,

where the integration path is a line from 0 to z in the complex t-plane. For the case m1<Re(μ)<m (m=1,2,3,), it is defined by

D z μ { f ( z ) } = d m d z m D z μ m { f ( z ) } = d m d z m { 1 Γ ( μ + m ) 0 z f ( t ) ( z t ) μ + m 1 d t } .

The extended Riemann-Liouville fractional derivative operator was defined by Özarslan and Özergin as follows.

Definition 8 ([18])

The extended Riemann-Liouville fractional derivative is defined as

D z μ , p { f ( z ) } = 1 Γ ( μ ) 0 z f(t) ( z t ) μ 1 exp ( p z 2 t ( z t ) ) dt,Re(μ)<0,Re(p)>0
(20)

and for m1<Re(μ)<m (m=1,2,3,)

D z μ , p { f ( z ) } = d m d z m D z μ m { f ( z ) } = d m d z m { 1 Γ ( μ + m ) 0 z f ( t ) ( z t ) μ + m 1 exp ( p z 2 t ( z t ) ) d t } ,

where the path of integration is a line from 0 to z in the complex t-plane. For the case p=0, we obtain the classical Riemann-Liouville fractional derivative operator.

We begin by the following theorem.

Theorem 9 Let p0, Re(μ)>Re(λ)>0, Re(α)>0, Re(β)>0. Then

D z λ μ , p { z λ 1 E α , β ( c ) ( z ) } = z μ 1 B ( λ , c λ ) Γ ( μ λ ) E α , β ( λ ; μ ) (z;p).

Proof Replacing μ by λμ in the definition of the extended fractional derivative operator (20), we get

D z λ μ , p { z λ 1 E α , β ( c ) ( z ) } = 1 Γ ( μ λ ) 0 z t λ 1 E α , β ( c ) ( t ) ( z t ) λ + μ 1 exp { p z 2 t ( z t ) } d t = z λ + μ 1 Γ ( μ λ ) 0 z t λ 1 E α , β ( c ) ( t ) ( 1 t z ) λ + μ 1 exp { p z 2 t ( z t ) } d t .
(21)

Taking u= t z in equation (21), we get

D z λ μ , p { z λ 1 E α , β ( c ) ( z ) } = z μ 1 Γ ( μ λ ) 0 1 u λ 1 ( 1 u ) λ + μ 1 exp { p u ( 1 u ) } E α , β ( c ) ( u z ) d u .
(22)

Comparing this result with equation (6), we get

D z λ μ , p { z λ 1 E α , β ( c ) ( z ) } = z μ 1 B ( λ , c λ ) Γ ( μ λ ) E α , β ( λ ; μ ) (z;p).

Whence the result. □

In the following theorem, we give the derivative properties of the extended Mittag-Leffler function.

Theorem 10 For the extended Mittag-Leffler function, we have the following derivative formula:

d n d z n { E α , β ( γ ; c ) ( z ; p ) } = ( c ) n E α , β + n α ( γ + n ; c + n ) (z;p),nN.
(23)

Proof Taking the derivative with respect to z in equation (6), we get

d d z { E α , β ( γ ; c ) ( z ; p ) } =c E α , β + α ( γ + 1 ; c + 1 ) (z;p).
(24)

Again taking the derivative with respect to z in equation (24), we get

d 2 d z 2 { E α , β ( γ ; c ) ( z ; p ) } =c(c+1) E α , β + 2 α ( γ + 2 ; c + 2 ) (z;p).
(25)

Continuing the repetition of this procedure n times, we get the desired result. □

Theorem 11 For the extended Mittag-Leffler function, the following differentiation formula holds:

d n d z n { z β 1 E α , β ( γ ; c ) ( λ z α ; p ) } = z β n 1 E α , β n ( γ ; c ) ( λ z α ; p ) .

Proof In equation (23), replace z by λ z α and multiply z β 1 , then taking the z-derivative n times, we get the result. □

Theorem 12 For the extended Mittag-Leffler function, the following differentiation formula holds:

d n d p n { E α , β ( γ ; c ) ( z ; p ) } = ( 1 ) n Γ ( γ n ) Γ ( c γ n ) Γ ( c ) Γ ( c 2 n ) Γ ( γ ) Γ ( c γ ) E α , β ( γ n ; c 2 n ) (z;p).

Proof Taking the p-derivative n times in equation (6), we get the result. □

4 Relations between the extended Mittag-Leffler function with Laguerre polynomial and Whittaker function

In this section, we give a representation of the extended Mittag-Leffler function in terms of Laguerre polynomials and Whittaker’s function.

Theorem 13 For the extended Mittag-Leffler function, we have

exp ( 2 p ) E α , β ( γ ; c ) ( z ; p ) = 1 B ( γ , c γ ) m , n , k = 0 L m ( p ) L n ( p ) ( c ) k Γ ( α k + β ) k ! z k B ( m + k + γ + 1 , n + c γ + 1 ) ,

where Re(c)>Re(γ)>0, Re(α)>0, Re(β)>0.

Proof We start by recalling the useful identity used in [18]

exp ( p t ( 1 t ) ) =exp(2p) m , n = 0 L n (p) L m (p) t m + 1 ( 1 t ) n + 1 ;0<t<1.
(26)

Using equation (26) in equation (6), we get

E α , β ( γ ; c ) ( z ; p ) = 1 B ( γ , c γ ) 0 1 t γ 1 ( 1 t ) c γ 1 exp ( 2 p ) × m , n = 0 L n ( p ) L m ( p ) t m + 1 ( 1 t ) n + 1 E α , β c ( t z ) d t .
(27)

Now, taking into account the series expansion of Prabhakar’s generalized Mittag-Leffler’s function E α , β c (tz) in equation (27), we have

E α , β ( γ ; c ) ( z ; p ) = exp ( 2 p ) B ( γ , c γ ) 0 1 t γ 1 ( 1 t ) c γ 1 × m , n = 0 L n ( p ) L m ( p ) t m + 1 ( 1 t ) n + 1 k = 0 ( c ) k ( t z ) k Γ ( α k + β ) k ! d t = exp ( 2 p ) B ( γ , c γ ) 0 1 t γ 1 ( 1 t ) c γ 1 × m , n , k = 0 L n ( p ) L m ( p ) ( c ) k Γ ( α k + β ) k ! t m + k + 1 ( 1 t ) n + 1 z k d t .
(28)

Interchanging the order of integration and summation in equation (28), which can be done under the assumptions of the theorem, we have

E α , β ( γ ; c ) ( z ; p ) = exp ( 2 p ) B ( γ , c γ ) m , n , k = 0 L n ( p ) L m ( p ) ( c ) k Γ ( α k + β ) k ! z k × B ( m + k + γ + 1 , n + c γ + 1 ) .
(29)

Multiplying both sides of equation (29) by exp(2p), we get the result. □

In the following theorem, we give the extended Mittag-Leffler function in terms of Whittaker’s function.

Theorem 14 For the extended Mittag-Leffler function we have

exp ( 3 p 2 ) E α , β ( γ ; c ) (z;p)= Γ ( c γ + 1 ) B ( γ , c γ ) m , k = 0 L m ( p ) ( c ) k Γ ( α k + β ) k ! p m + k + γ 1 2 W γ 2 c m 1 2 , m + k + γ 2 (p).

Proof Considering the following equality:

exp ( p t ( 1 t ) ) =exp ( p 1 t ) exp ( p t ) ,

and using the generating function of the Laguerre polynomials, we get

exp ( p t ( 1 t ) ) =exp(p)exp ( p t ) (1t) m = 0 L m (p) t m .
(30)

Taking equation (30) into account in equation (6), we have

E α , β ( γ ; c ) ( z ; p ) = 1 B ( γ , c γ ) 0 1 t γ 1 ( 1 t ) c γ 1 e p t ( 1 t ) E α , β ( c ) ( t z ) d t = 1 B ( γ , c γ ) 0 1 t γ 1 ( 1 t ) c γ 1 exp ( p ) exp ( p t ) × ( 1 t ) m = 0 L m ( p ) t m E α , β ( c ) ( t z ) d t .
(31)

By use of Prabhakar’s generalized Mittag-Leffler function E α , β ( c ) (tz) in equation (31), we get

E α , β ( γ ; c ) ( z ; p ) = exp ( p ) B ( γ , c γ ) 0 1 t γ 1 ( 1 t ) c γ exp ( p t ) m = 0 L m ( p ) t m k = 0 ( c ) k t k z k Γ ( α k + β ) k ! d t .
(32)

Interchanging the order of summation and integration in equation (32), we get

E α , β ( γ ; c ) ( z ; p ) = exp ( p ) B ( γ , c γ ) m , k = 0 L m ( p ) ( c ) k z k Γ ( α k + β ) k ! 0 1 t m + k + γ 1 ( 1 t ) c γ exp ( p t ) d t .
(33)

Finally, using the following integral representation [19]:

0 1 t μ 1 ( 1 t ) ν 1 exp ( p t ) d t = Γ ( ν ) p μ 1 2 exp ( p 2 ) W 1 μ 2 ν 2 , μ 2 ( p ) ( Re ( ν ) > 0 , Re ( p ) > 0 ) ,

in equation (33), we get the result. □

References

  1. Camargo RF, Capelas de Oliveira E, Vaz J: On the generalized Mittag-Leffler function and its application in a fractional telegraph equation. Math. Phys. Anal. Geom. 2012,15(1):1-16. 10.1007/s11040-011-9100-8

    Article  MathSciNet  MATH  Google Scholar 

  2. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, New York; 1993.

    MATH  Google Scholar 

  3. Haubold HJ, Mathai AM, Saxena RK: Mittag-Leffler functions and their applications. J. Appl. Math. 2011., 2011: Article ID 298628 10.1155/2011/298628

    Google Scholar 

  4. Soubhia AL, Camargo RF, Capelas de Oliveira E, Vaz J: Theorem for series in three-parameter Mittag-Leffler function. Fract. Calc. Appl. Anal. 2010, 13: 1-12.

    MathSciNet  MATH  Google Scholar 

  5. Mittag-Leffler GM: Une généralisation de l’intégrale de Laplace-Abel. C. R. Acad. Sci., Ser. II 1903, 137: 537-539.

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  7. Mittag-Leffler GM: Sur la représentation analytique d’une fonction monogéne (cinquiéme note). Acta Math. 1905,29(1):101-181. 10.1007/BF02403200

    Article  MathSciNet  MATH  Google Scholar 

  8. Wiman A:Über der Fundamentalsatz in der Theorie der Funktionen E α (x). Acta Math. 1905, 29: 191-201. 10.1007/BF02403202

    Article  MathSciNet  MATH  Google Scholar 

  9. Wiman A:Über die Nullstellen der Funktionen E α (x). Acta Math. 1905, 29: 217-234. 10.1007/BF02403204

    Article  MathSciNet  MATH  Google Scholar 

  10. Agarwal RP: A propos d’une note de M. Pierre Humbert. C. R. Séances Acad. Sci. 1953, 236: 2031-2032.

    MathSciNet  MATH  Google Scholar 

  11. Humbert P, Agarwal RP: Sur la fonction de Mittag-Leffler et quelques unes de ses generalizations. Bull. Sci. Math., Ser. II 1953, 77: 180-185.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  13. Kurulay M, Bayram M: Some properties of the Mittag-Leffler functions and their relation with the Wright function. Adv. Differ. Equ. 2012., 2012: Article ID 178

    Google Scholar 

  14. Mainardi, F: On some properties of the Mittag-Leffler function E α (tz), completely monotone for t > 0 with 0 < α < 1. arXiv: http://arxiv.org/abs/arXiv:1305.0161

  15. Chaudhry MA, Qadir A, Srivastava HM, Paris RB: Extended hypergeometric and confluent hypergeometric functions. Appl. Math. Comput. 2004, 159: 589-602. 10.1016/j.amc.2003.09.017

    Article  MathSciNet  MATH  Google Scholar 

  16. Chaudhry MA, Zubair SM: On a Class of Incomplete Gamma Functions with Applications. 2002.

    MATH  Google Scholar 

  17. Srivastava HM, Manocha HL: A Treatise on Generating Functions. Halsted, New York; 1984. (Ellis Horwood, Chichester)

    MATH  Google Scholar 

  18. Özarslan MA, Özergin E: Some generating relations for extended hypergeometric functions via generalized fractional derivative operator. Math. Comput. Model. 2010, 52: 1825-1833. 10.1016/j.mcm.2010.07.011

    Article  MathSciNet  MATH  Google Scholar 

  19. Özarslan MA: Some remarks on extended hypergeometric, extended confluent hypergeometric and extended Appell’s functions. J. Comput. Anal. Appl. 2012,14(6):1148-1153.

    MathSciNet  MATH  Google Scholar 

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Correspondence to Banu Yılmaz.

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Özarslan, M.A., Yılmaz, B. The extended Mittag-Leffler function and its properties. J Inequal Appl 2014, 85 (2014). https://doi.org/10.1186/1029-242X-2014-85

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