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Some results on the generalized Mittag-Leffler function operator

Abstract

This paper is devoted to the study of a generalized Mittag-Leffler function operator introduced by Shukla and Prajapati (J. Math. Anal. Appl. 336:797-811, 2007). Laplace and Mellin transforms of this operator are investigated in this paper. Some special cases of the established results are also deduced as corollaries. The results obtained are useful where the Mittag-Leffler function occurs naturally.

MSC:33E12, 26A33, 44A45.

1 Introduction

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

E α (z)= n = 0 z n Γ ( α n + 1 ) ( α C ; Re ( α ) > 0 ) .
(1.1)

A generalization of (1.1) was given by Wiman [3] in 1905 in the form

E α , β (z)= n = 0 z n Γ ( α n + β ) ( α , β C ; Re ( α ) > 0 , Re ( β ) > 0 ) .
(1.2)

In 1971, in connection with the solution of certain singular integral equations, a further interesting and useful generalization of (1.2) was introduced by Prabhakar [4] in the form

(1.3)

where ( γ ) n is the Pochhammer symbol defined by

( γ ) 0 =1; ( γ ) n =γ(γ+1)(γ+n1)= Γ ( γ + n ) Γ ( γ ) ;γ0.
(1.4)

A generalization of (1.3) is given by Shukla and Prajapati [5] in the following form:

E α , β γ , q (z)= n = 0 ( γ ) q n z n Γ ( α n + β ) n ! ,
(1.5)

where (α,β,γC; Re(α)>0, Re(β)>0, Re(γ)>0, q(0,1)N).

The above generalization studied by Shukla and Prajapati is shown in a series of papers [58]. A generalization of Mittag-Leffler functions defined by (1.3) and (1.5) is introduced and studied by Srivastava and Tomovski [9] in the form

E α , β γ , κ (z)= n = 0 ( γ ) n κ z n Γ ( α n + β ) n ! ,
(1.6)

where α,β,γ,κC; Re(α)=Re(κ)1>0, Re(β)>0, Re(γ)>0, Re(κ)>0 and the Pochhammer symbol for λ,μC is defined by

( λ ) μ = Γ ( λ + μ ) Γ ( λ ) ={ 1 ( μ = 0 , λ C { 0 } ) ; λ ( λ + 1 ) ( λ + n 1 ) , μ = n N ; λ C .
(1.7)

The object of this paper is to derive the Laplace and Mellin transforms of the following integral operator associated with the generalized Mittag-Leffler function, defined by Shukla and Prajapati [6] as well as Srivastava and Tomovski [9], in the next sections:

( E α , β , ω ; 0 + γ , q f ) (x)= 0 x ( x t ) β 1 E α , β γ , q [ w ( x t ) α ] f(t)dt,
(1.8)

where α,β,γC; Re(α)>0, Re(β)>0, Re(γ)>0; q(0,1)N.

Shukla and Prajapati [6] have shown that the Mellin-Barnes integral for the function defined by (1.5) is given by

E α , β γ , q (z)= 1 2 π i Γ ( γ ) i i Γ ( ξ ) Γ ( γ + q ξ ) Γ ( α ξ + β ) ( z ) ξ dξ.
(1.9)

As γ0, then by virtue of the limit formula,

lim γ 0 E α , β γ (z)= 1 Γ ( β )
(1.10)

reduces to the familiar Reimann-Liouville fractional integral

( I 0 + α f ) (x)= 1 Γ ( α ) 0 x ( x t ) α 1 f(t)dt ( α C , Re ( α ) > 0 ) .
(1.11)

In the following, we will use the representation of a high transcendental function in terms of the so-called H-function defined as [10]

H p , q m , n [ z ( a 1 , A 1 ) , , ( a p , A p ) ( b 1 , B 1 ) , , ( b q , B q ) ] = 1 2 π i L θ(ξ) z ξ dξ,
(1.12)

where

θ(ξ)= [ j = 1 m Γ ( b j + B j ξ ) ] [ j = 1 n Γ ( 1 a j A j ξ ) ] [ j = m + 1 q Γ ( 1 b j B j ξ ) ] [ j = n + 1 p Γ ( a j + A j ξ ) ] ,
(1.13)

and an empty product is always interpreted as unity; m,n,p,q N 0 with 0np, 1mq, A i , B j R + , a i , b j R or (i=1,,p; j=1,,q) such that

A i ( b j +k) B j ( a i l1)(k,l N 0 ;i=1,,n;j=1,,m).
(1.14)

The contour L is the path of integration in the complex ξ-plane running from γi to γ+i for some real number γ.

2 Mellin transform of the operator (1.8)

Theorem 2.1 It is shown here that

M { ( E α , β , ω ; 0 + γ , q f ) ( x ) ; s } = 1 Γ ( γ ) Γ ( 1 s ) H 1 , 2 2 , 1 [ w t α ( 1 γ , q ) ( 0 , 1 ) , ( 1 s β , α ) ] M { t β f ( t ) ; s } ,
(2.1)

where (Re(α)>0, Re(β)>0, Re(γ)>0); q(0,1)N, Re(1sβ)>0and H 1 , 2 2 , 1 ()is the H-function defined by (1.12).

Proof

Mellin transform is defined as

M { f ( x ) ; s } = 0 x s 1 f(x)dx.

Therefore, we have

M { ( E α , β , ω ; 0 + γ , q f ) ( x ) ; s } = 0 x s 1 0 x ( x t ) β 1 E α , β γ , q [ ω ( x t ) α ] f(t)dtdx.

Interchanging the order of integration, which is permissible under the conditions given in Theorem 2.1, we find that

(2.2)

If we consider x=t+u in the r.h.s. of (2.2), we get

M { ( E α , β , ω ; 0 + γ , q f ) ( x ) ; s } = 0 f(t)dt t ( t + u ) s 1 u β 1 E α , β γ , q [ ω u α ] du.
(2.3)

To evaluate the u-integral, we express the Mittag-Leffler function in terms of its Mellin-Barnes contour integral by means of the formula (1.9), then the above expression transforms into the form

M { ( E α , β , ω ; 0 + γ , q f ) ( x ) ; s } = 0 f ( t ) d t 1 2 π i Γ ( γ ) i i Γ ( ξ ) Γ ( γ + q ξ ) Γ ( α ξ + β ) ( ω ) ξ 0 ( t + u ) s 1 u β + α ξ 1 d u d ξ .
(2.4)

If the u-integral is evaluated with the help of the formula

0 x ν 1 ( x + a ) ρ dx= Γ ( ν ) Γ ( ρ ν ) Γ ( ρ ) ;Re(ρ)>Re(ν)>0,
(2.5)

then after some simplification, it is seen that the right-hand side of above equation (2.4) simplifies to

(2.6)

which, on being interpreted by the definition of H-function (1.12), yields the desired result. □

For q=1, Theorem 2.1 reduces to the following corollary.

Corollary 2.1 The following result holds:

M { ( E α , β , ω ; 0 + γ f ) ( x ) ; s } = 1 Γ ( γ ) Γ ( 1 s ) H 1 , 2 2 , 1 [ w t α ( 1 γ , 1 ) ( 0 , 1 ) , ( 1 s β , α ) ] M { t β f ( t ) ; s } ,
(2.7)

where (Re(α)>0, Re(β)>0, Re(γ)>0); Re(1sβ)>0and H 1 , 2 2 , 1 ()is the H-function defined by (1.12).

Theorem 2.2 It is shown here that

M { ( E α , β , ω ; 0 + γ , κ f ) ( x ) ; s } = 1 Γ ( γ ) Γ ( 1 s ) H 1 , 2 2 , 1 [ w t α ( 1 γ , κ ) ( 0 , 1 ) , ( 1 s β , α ) ] M { t β f ( t ) ; s } ,
(2.8)

whereRe(α)=Re(κ)1>0, Re(β)>0, Re(γ)>0, Re(κ)>0, Re(1sβ)>0and H 1 , 2 2 , 1 ()is the H-function defined by (1.12).

3 Laplace transform of the operator (1.8)

Theorem 3.1 The following result holds:

L { ( E α , β , ω ; 0 + γ , q f ) ( x ) ; p } = 1 Γ ( γ ) p 1 β ψ 0 [ ( γ , q ) ; ω / p α ; ] F(p),
(3.1)

where (Re(α),Re(β),Re(γ)>0); Re(p)> | ω | 1 / Re ( α ) andF(p)is the Laplace transform off(t)defined by

L { f ( t ) ; p } =F(p)= 0 e p t f(t)dt,
(3.2)

whereRe(p)>0and the integral is convergent.

Proof By virtue of the definitions (1.8) and (3.2), it follows that

L { ( E α , β , ω ; 0 + γ , q f ) ( x ) ; p } = 0 e p x 0 x ( x t ) β 1 E α , β γ , q [ ω ( x t ) α ] f(t)dtdx.

Interchanging the order of integration, which is permissible under the conditions given in Theorem 3.1, we find that

L { ( E α , β , ω ; 0 + γ , q f ) ( x ) ; p } = 0 f(t)dt t e p x ( x t ) β 1 E α , β γ , q [ ω ( x t ) α ] dx.

If we consider x=t+u, we obtain

L { ( E α , β , ω ; 0 + γ , q f ) ( x ) ; p } = 0 e p t f(t)dt 0 e p u u β 1 E α , β γ , q [ ω u α ] du.

On making use of the series definition (1.5), the above expression becomes

and F(p) is the Laplace transform of f(t). □

For q=1, Theorem 3.1 reduces to the following corollary.

Corollary 3.1 The following result holds:

Ł { ( E α , β , ω ; 0 + γ f ) ( x ) ; p } = p β ( 1 a p α ) γ F(p),
(3.3)

where (Re(α)>0, Re(β)>0, Re(γ)>0); Re(p)>0and the integral operator is the one discussed by Prabhakar[4]defined by (1.10).

Theorem 3.2 The following result holds:

L { ( E α , β , ω ; 0 + γ , κ f ) ( x ) ; p } = 1 Γ ( γ ) p 1 β ψ 0 [ ( γ , κ ) ; ω / p α ; ] F(p),
(3.4)

whereRe(α)=Re(κ)1>0, Re(β)>0, Re(γ)>0, Re(κ)>0; Re(p)> | ω | 1 / Re ( α ) andF(p)is the Laplace transform off(t)defined by (3.2).

References

  1. Mittag-Leffler GM: Sur la nouvelle function. C. R. Math. Acad. Sci. Paris 1903, 137: 554–558.

    Google Scholar 

  2. Mittag-Leffler GM: Sur la representation analytique d’une function monogene (cinquieme note). Acta Math. 1905, 29: 101–181. 10.1007/BF02403200

    Article  MATH  MathSciNet  Google Scholar 

  3. Wiman A: Uber den Fundamental Salz in der Theorie der Funktionen. Acta Math. 1905, 29: 191–201. 10.1007/BF02403202

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  5. Shukla AK, Prajapati JC: On a generalization of Mittag-Leffler function and its properties. J. Math. Anal. Appl. 2007, 336: 797–811. 10.1016/j.jmaa.2007.03.018

    Article  MATH  MathSciNet  Google Scholar 

  6. Shukla AK, Prajapati JC: On a generalization of Mittag-Leffler type function and generalized integral operator. Math. Sci. Res. J. 2008, 12(12):283–290.

    MATH  MathSciNet  Google Scholar 

  7. Shukla AK, Prajapati JC: On a recurrence relation of generalized Mittag-Leffler function. Surv. Appl. Math. 2009, 4: 133–138.

    MATH  MathSciNet  Google Scholar 

  8. Shukla AK, Prajapati JC: Some remarks on generalized Mittag-Leffler function. Proyecciones 2009, 28(1):27–34.

    MATH  MathSciNet  Google Scholar 

  9. Srivastava HM, Tomovski Z: Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel. Appl. Math. Comput. 2009, 211: 198–210. 10.1016/j.amc.2009.01.055

    Article  MATH  MathSciNet  Google Scholar 

  10. Mathai AM, Saxena RK, Haubold HJ: The H-Function: Theory and Applications. Springer, New York; 2010.

    Book  Google Scholar 

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Prajapati, J., Jana, R., Saxena, R. et al. Some results on the generalized Mittag-Leffler function operator. J Inequal Appl 2013, 33 (2013). https://doi.org/10.1186/1029-242X-2013-33

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Keywords

  • generalized Mittag-Leffler function
  • Laplace transform
  • Mellin transform
  • H-function
  • generalized Wright function