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J. Comp. & Math. Sci. Vol.3 (5), 536-541 (2012)

Certain Families of Generalized Mittag-Leffler Functions and their Integral Representation HARISH NAGAR and NARESH MENARIA Department of Mathematics, Mewar University, Chittorgarh, INDIA. (Received on: October 8, 2012) ABSTRACT In this paper integral representation and some other results are established for some families of Mittag-Leffler function denoted , by E , z and E , ι , β , ; z which are introduced and studied by Shukla and Prjapati respectively.

and Saxena and Nishimoto

Keywords: Generalization of Mittag-Leffler function, integral representation and binomial expression. 2010 Mathematical Subject Classification: 26A33,33C65.

1. INTRODUCTION AND PRELIMINARIES The entire function of the form ∑

(1.1)

Where Îą C ,Re (Îą) > 0 z C, defines the Mittag-Leffler function5. A Generalization of (1.1) in the form

, ∑

A Generalization of (1.2) is introduced in terms of series representation by Prabhakar10 as

,

1.3

k !

Where ι, β, γ C; Re (ι) > 0, Re (β) > 0 , Re (γ) > 0, z C and is the well-known Pochhammer symbol. A Generalization of (1.3) is introduced by Shukla and Prajapati2 in the following form

(1.2)

Where ι, β C; Re (ι) > 0 , Re (β) > 0, z C is defined and studied by Wiman3.

, ,

n !

Journal of Computer and Mathematical Sciences Vol. 3, Issue 5, 31 October, 2012 Pages (498-556)

1.4

537

Harish Nagar, et al., J. Comp. & Math. Sci. Vol.3 (5), 536-541 (2012)

where Îą, β, Îł C; Re (Îą) > 0 , Re (β) > 0, Re (Îł) > 0, β > Îą >0 and q ∈ (0, 1) U N and denotes the

generalized Pochhammer symbol [4] which in particular reduces to �

if q . Another Generalization of Mittag-Leffler function defined in (1.3) and (1.4) was introduced and studied by Srivastava and Tomovski6 in the form

, ,

n !

1.5

Where Îą, β, Îł , " C, Re(Îą) > max{0,Re(K)-1}, Re (β) > 0 , Re (Îł) > 0 Re (K) > 0 . A further extension of both the MittagLeffler function defined in (1.5) and multiindex Mittag-Leffler function defined by Kriyakova7,8 was recently introduced and studied by Saxena and Nishimoto9 in the form , , , ; , , , ‌ , , ;

� Γ

r !

1.6

Where r, , , γ C; #$% & ' 0, #$% & ' 0, #$ ' 0, ) 1,2, ‌ , . When m=1 (1.6) reduces to (1.5).It is interesting to observe that for γ=k=1, (1.6) yields the multi-index Mittag-Leffler function, defined by Kriyakova7 in the form

7

1 , , , ; ŕł•

� Γ

# $

% & n !

Where Îą) , β) C; Re%Îą) & ' 0, #$%β) & ' 0, j 1,2, ‌ m . 2. INTEGRAL REPRESENTATION ,,OF 3 *,+ 4 In this section we obtained integral /,0 representation of the function E ., z which is introduced by Shukla and Prajapati2. Theorem 2.1 If Îą, β, Îł C; Re (Îą) > 0 , Re (β) > 0 , Re (Îł) > 0, β > Îą >0 and q ∈ (0, 1) U N then 4

1 5 4 1 5 94 6 Îł $ % 6 % $ 5 Îł 5 qn

E , z 3 ,

Proof. To prove the theorem 2.1we denote its right hand side by 7 i.e.

1 ! 1 !

' " Îł $ " $ ! Îł ! qn

Now using binomial expression 4 1 5 4 : 3 6 ; $ % 6 % $ 5 ; 5 <%

; 94 %!

Now interchanging the order of integration and summation we get ஶ

ଵ

௥ŕ­€଴

௥ ௥ ŕ°Šା௤௡ିଵ 1 n ! Îł Îł qn ଵ

ŕ°ˆ௡ାŕ°‰ିŕ°Šି௤௡ିଵ

8 8 ! 8 8

Using generalized Pochhammer symbol and solving we get

Journal of Computer and Mathematical Sciences Vol. 3, Issue 5, 31 October, 2012 Pages (498-556)

଴

Harish Nagar, et al., J. Comp. & Math. Sci. Vol.3 (5), 536-541 (2012)

7

538

n !

/,=

7 E ., z

3. INTEGRAL REPRESENTATION OF 3,,> %9? , :? @,A ; 4<

In this section we obtained integral representation of the function E/,B %ι) , β) ,C ; z< which is introduced by Saxena and Nishimoto9

Theorem 3.1 If n, , , Îł C; #$% & ' 0, #$% & ' 0, #$ ' 0, ) 1,2, ‌ , then / DB > 1 8 > D DB 1 8 / B> E/,B %Îą) , β) ,C ; z< = âˆ?E

Γ % @ & r 8 rk Proof. To prove the theorem 3.1 we denote its right hand side by 7F i.e. / DB > 1 8 > D DB 1 8 / 7F = B> �E

Γ % @ & r 8 rk Now using binomial expression

> / DB 1 8 > D DB 7F = C E D B> @! � Γ % @ & r 8 rk

Now interchanging the order of integration and summation we have

7F

r ! �E Γ % @ & @ 8 @

= > / DB 1 8 > D DB B>

Îł @ @ 8 @ & @ 8 @ Îł @ @ 8 @ Using generalized Pochhammer symbol and solving we get

7F r ! �E Γ % @ &

7F

r ! �E Γ % @

7F E/,B %ι) , β) ,C ; z<

4. CERTAIN FAMILIES OF GENERALIZED MITTAG-LEFFLER FUNCTIONS In this section we obtain some results related to generalized Mittag-Leffler functions /,= denoted by E ., z and E/,B %ι) , β) ,C ; z< which are introduced and studied by Shukla and Journal of Computer and Mathematical Sciences Vol. 3, Issue 5, 31 October, 2012 Pages (498-556)

539

Harish Nagar, et al., J. Comp. & Math. Sci. Vol.3 (5), 536-541 (2012)

Prjapati2 and Saxena and Nishimoto9 respectively. To obtain the results we use binomial expression and gamma function. The results follows from the following theorems. Theorem 4.1.If Îą, β, Îł C; Re (Îą) > 0 , Re (β) > 0 , Re (Îł) > 0, β > Îą >0 and q ∈ (0, 1) U N then

F 1 8 G , 4.1 ,

n ! k !

Proof . To prove the result in 4.1 we denote its right hand side by 7H i.e.

7H

7H

F 1 8 G n ! k !

F 1 8 G k! n !

Now using binomial expression

7H

n !

7H , ,

Theorem 4.2.If @, , G, #$% & ' 0, #$% & ' 0, #$ ' 0, ) 1, ‌ , , then

E/,B %ι) , β) ,C ; z<

F 1 8 G n ! r ! �E Γ % @ &

Proof . To prove the result in 4.2 we denote its right hand side by 7I i.e.

7I

7I

F 1 8 G n ! r ! �E Γ % @ &

F 1 8 G n! �E Γ % @ &r !

Journal of Computer and Mathematical Sciences Vol. 3, Issue 5, 31 October, 2012 Pages (498-556)

4.2

Harish Nagar, et al., J. Comp. & Math. Sci. Vol.3 (5), 536-541 (2012)

540

Now using binomial expression

7I E � Γ % @ &r !

7I E/,B %ι) , β) ,C ; z<

Theorem 4.3 If Îą, β, Îł C, a,b>0; Re (Îą) > 0, Re (β) > 0, Re (Îł) > 0, β > Îą >0 and q ∈ (0, 1) U N then , ,

, H E $ JK =

BH n ! Γ m 1 Γ Îąn β H

4.3

Proof . To prove the result in 4.3 we denote its right hand side by 7L i.e.

7L =

, H E $ JK BH n ! Γ m 1 Γ Îąn β H

E 7L = H E $ JK BH n ! Γ Îąn β Γ m

Now using definition of Gamma function we get

7L

E Γ m n ! Γ Îąn β Γ m E

Using (1.4) above equation immediately leads to 7L , ,

Theorem 4.4 If r, , , γ C; #$% & ' 0, #$% & ' 0, #$ ' 0, ) 1,2, ‌ , then

, % , ,E ; < =

, H E $ JK BH �E Γ % @ & r! HΓ m 1

Proof To prove the result in 4.4 we denote its right hand side by 7M i.e

7M =

, H E $ JK

�E Γ % @ & r! HΓ m 1

BH

Journal of Computer and Mathematical Sciences Vol. 3, Issue 5, 31 October, 2012 Pages (498-556)

4.4

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Harish Nagar, et al., J. Comp. & Math. Sci. Vol.3 (5), 536-541 (2012)

7M

E JK E = $ H BH �E Γ @ r! Γ m

Now using (1.6) and Gama function we get 7M , % , ,E ; < REFERENCES 1. A. K. Shukla and J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties. J. Math. Anal. Appl. 336, pp.797-811 (2007). 2. A.Wiman,Uber den Fundamental satz in der the theorie der functionen H , Acta Math. 29, 191-201 (1905). 3. E. D. Rainville, Special Functions, Macmillan- New York, (1960). 4. G. M. Mittag-Leffler, Sur la nouvelle fonction EÎą(x), C. R. Acad. Sci. Paris 137, 554-558 (1903). 5. H.M. Srivastava, and Tomovski, Z. : Fractional Calculus with an integral operator containing a generalized Mittag-Leffler function in the Kernel.

Appl. Math. Comput. 21,198-210 (2009). 6. V.S. Kiryakova: Multi-index MittagLeffler functions related to GelfondLeontier operators and Laplace type integral transforms, Fract. Calc. Appl. Anal. 2,445-462 (1999). 7. V.S. Kiryakova: Muti-index MittagLeffler function and relations to generalized fractional calculus. J. Comput. Appl. Math. 118, 214-249 (2000). 8. R.K. Saxena, and K. Nishimoto: NFractional Calculus of Generalized Mittag-Leffler functions, J. Fract. Calc., 37, 43-52 (2010). 9. T. R. Prabhakar, A singular integral equation with a generalized MittagLeffler function in the kernel, Yokohama Math. J. 19,7-15 (1971).

Journal of Computer and Mathematical Sciences Vol. 3, Issue 5, 31 October, 2012 Pages (498-556)

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