Research Article Generalized Mittag-Leffler Function ...downloads.hindawi.com/journals/jmath/2013/821762.pdf · function; its various properties including di erentiation, Laplace,Beta,andMellintransforms,andgeneralizedhyper-geometric

Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 821762 5 pageshttpdxdoiorg1011552013821762

Research ArticleGeneralized Mittag-Leffler Function Associated withWeyl Fractional Calculus Operators

Ahmad Faraj1 Tariq Salim1 Safaa Sadek2 and Jamal Ismail2

1 Department of Mathematics Al-Azhar University-Gaza PO Box 1277 Gaza Palestine2 Department of Mathematics College of Girls Ain Shams University Cairo Egypt

Correspondence should be addressed to Tariq Salim trsalimyahoocom

Received 8 January 2013 Revised 1 April 2013 Accepted 18 April 2013

Academic Editor Josefa Linares-Perez

Copyright copy 2013 Ahmad Faraj et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is devoted to study further properties of generalized Mittag-Leffler function 119864120574120575119902

120572120573119901associated with Weyl fractional

integral and differential operators A new integral operatorE120574120575119902120572120573119901119908infin

depending onWeyl fractional integral operator and containing119864120574120575119902

120572120573119901(119911) in its kernel is defined and studied namely its boundedness Also composition ofWeyl fractional integral and differential

operators with the new operator E120574120575119902120572120573119901119908infin

is established

1 Introduction

In 1903 the Swedish mathematician Mittag-Leffler [1] intro-duced the function 119864120572(119911) as

119864120572 (119911) =

infin

sum

119899=0

119911119899

Γ (120572119899 + 1) (1)

where 119911 isin C and Γ(119904) is the gamma function 120572 ge 0During the last century and due to its involvement in the

problems of physics engineering and applied sciences manyauthors defined and studied in their research papers differ-ent generalizations of Mittag-Leffler type function namely119864120572120573(119911) introduced byWiman [2]119864120574

120572120573(119911) stated by Prabhakar

[3] 119864120574119902120572120573

(119911) defined and studied by Shukla and Prajapati [4]and 119864

120574120575

120572120573(119911) investigated by Salim and Faraj [5]

Prabhakar studied some properties of generalizedMittag-Leffler type-function 119864

120574

120572120573(119911) and the fractional integral oper-

ator

(E120574

120572120573119908119886+120593) (119909) = int

119909

119886

(119909 minus 119905)120573minus1

119864120574

120572120573[119908(119909 minus 119905)

120572] 120593 (119905) 119889119905

(2)

containing 119864120574

120572120573(119911) in the kernel and applied the result

obtained to prove the existence and uniqueness of thesolution of corresponding integral equation of the first kind

Moreover Kilbas et al [6] devoted themselves to furtherinvestigation of 119864120574

120572120573(119911) and the integral operator defined in

(2) They established integral representation differentiationand integration properties of 119864

120574

120572120573(119911) and formulas of its

Riemann-Liouville fractional integral and differential oper-ators For more results and conclusions one can refer to thework of Srivastava and Tomovski [7]

Recently Salim and Faraj [5] introduced a new general-ization of Mittag-Leffler-type function as

119864120574120575119902

120572120573119901(119911) =

infin

sum

119899=0

(120574)119902119899119911119899

Γ (120572119899 + 120573) (120575)119901119899

(3)

where

119911 120572 120573 120574 120575 isin C min Re (120572) Re (120573) Re (120574) Re (120575) gt 0

119901 119902 gt 0

(4)

Equation (3) is just a generalized formula of Mittag-Lefflerfunction its various properties including differentiationLaplace Beta andMellin transforms and generalized hyper-geometric series form and its relationship with other type ofspecial functions were investigated and established

2 Journal of Mathematics

On the other hand Salim and Faraj in their research paperdefined and studied an integral operator E120574120575119902

120572120573119901119908119886+as

(E120574120575119902

120572120573119901119908119886+120593) (119909) = int

119909

119886

(119909 minus 119905)120573minus1

119864120574120575119902

120572120573119901[119908(119909 minus 119905)

120572] 120593 (119905) 119889119905

(5)

containing 119864120574120575119902

120572120573119901(119911) in the kernel Also composition of

Riemann-Liouville fractional integral and differential opera-tors with the integral operator defined in (5) was established

This paper is devoted for the study of further propertiesof the generalizedMittag-Leffler function119864

120574120575119902

120572120573119901(119911) defined in

(3) with another type of fractional calculus operators calledWeyl fractional integral and differential operators written as

(119868120582

minus120593) (119909) =

1

Γ (120582)int

infin

119909

(119905 minus 119909)120582minus1

120593 (119905) 119889119905 (6)

(119863120582

minus120593) (119909)

= (minus1)119898(

119889

119889119909)

1198981

Γ (119898 minus 120582)int

infin

119909

(119905 minus 119909)119898minus120582minus1

120593 (119905) 119889119905

(7)

The last definition can be written in the form

(119863120582

minus120593) (119909) = (minus1)

119898(

119889

119889119909)

119898

(119868119898minus120582

minus120593) (119909) (8)

Precisely the authors investigate the basic properties of Weylfractional integral and differential operator with generalizedMittag-Leffler function 119864

120574120575119902

120572120573119901(119911) moreover a new integral

operator depending onWeyl fractional integral operator andcontaining 119864

120574120575119902

120572120573119901(119911) in its kernel is established as

(E120574120575119902

120572120573119901119908infin120593) (119908) = int

infin

119909

(119905 minus 119909)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

(9)

The condition of boundedness of the integral operator (9)is discussed and stated in the space 119871(119886infin) of Lebesgue-measurable functions on (119886infin)

119871 (119886infin) = 119892 (119909) 1003817100381710038171003817119892

10038171003817100381710038171= int

infin

119886

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909 lt infin (10)

Also composition of Weyl fractional integration and differ-entiation with the operator defined in (9) is established

Throughout this paper we need the followingwell-knownfacts and rules

(i) Fubinirsquos theorem (Dirichlet formula) [8]

int

119887

119886

119889119909int

119909

119886

119891 (119909 119905) 119889119905 = int

119887

119886

119889119905int

119887

119905

119891 (119909 119905) 119889119909 (11)

119889

119889119909int

119909

119886

ℎ (119909 119905) 119889119905 = [int

119909

119886

120597

120597119909ℎ (119909 119905) 119889119905] + ℎ (119909 119909) (12)

(ii) The Riemann-Liouville fractional integral [8]

(119868120582

119886+120593) (119909) =

1

Γ (120582)int

119909

119886

(119909 minus 119905)120582minus1

120593 (119905) 119889119905

(120572 isin CRe (120572) gt 0)

(13)

(iii) The Riemann-Liouville fractional derivative [8]

(119863120582

119886+120593) (119909) = (

119889

119889119909)

119899

(119868119899minus120582

119886+120593) (119909) 119899 = [Re (120572)] + 1 (14)

(iv) Beta transform (Sneddon [9])

119861 119891 (119911) 119886 119887 = int

1

0

119911119886minus1

(1 minus 119911)119887119891 (119911) 119889119911 (15)

where Re(119886) gt 0 Re(119887) gt 0

(v) The Beta function is written as

120573 (120572 120573) =Γ (120572) Γ (120573)

Γ (120572 + 120573) (16)

(vi) The difference property of the Gamma function is

Γ (120572 + 1) = 120572Γ (120572) (17)

2 Further Properties of Weyl FractionalIntegral Related to Mittag-Leffler Function

In this section we consider composition of Weyl fractionalintegral and derivative (6) and (7) with generalized Mittag-Leffler function 119864

120574120575119902

120572120573119901(119911) defined in (3)

Theorem 1 Let 120572 120573 120574 120575 120582 119908 isin CminRe(120572) Re(120573) Re(120574)Re(120575) Re(120582) gt 0 and 119901 119902 gt 0 then

119868120582

minus[119905minus120582minus120573

119864120574120575119902

120572120573119901(119908119905minus120572)] (119909) = 119909

minus120573119864120574120575119902

120572120573+120582119901(119908119909minus120572) (18)

Proof

119868120582


119864120574120575119902

120572120573119901(119908119905minus120572)] (119909)

=1

Γ (120582)int

infin

119909

(119905 minus 119909)120582minus1

119905minus120582minus120573

times

infin

sum

119899=0

(120574)119902119899119908119899119905minus120572119899

(120575)119901119899Γ (120572119899 + 120573)119889119905

=1

Γ (120582)

infin

sum

119899=0

(120574)119902119899119908119899

(120575)119901119899Γ (120572119899 + 120573)

times int

infin

119909

(119905 minus 119909)120582minus1

119905minus120582minus120573

119905minus120572119889119905

(19)

Journal of Mathematics 3

Let 119906 = (119905 minus 119909)119905 then

119868120582


119864120574120575119902

120572120573119901(119908119905minus120572

)] (119909)

=1

Γ (120582)

infin

sum

119899=0

(120574)119902119899119908119899119909minus120573minus120572119899

(120575)119901119899Γ (120572119899 + 120573)

times int

1

0

119906120582minus1

(1 minus 119906)120572119899+120573minus1

119889119906

=1

Γ (120582)

infin

sum

119899=0

(120574)119902119899119908119899119909minus120573minus120572119899

(120575)119901119899Γ (120572119899 + 120573)119861 (120582 120572119899 + 120573)

= 119909minus120573

119864120574120575119902

120572120573+120582119901(119908119909minus120572)

(20)

Theorem2 Let 120572 120573 120574 120575 120582119908 isin CminRe(120572)Re(120573)Re(120574)Re(120575) Re(120582) gt 0 Re(120573) gt [Re(120582)] + 1 and 119901 119902 gt 0 then

119863120582

minus[119905120582minus120573

119864120574120575119902

120572120573119901(119908119905minus120572)] (119909) = 119909

minus120573119864120574120575

120572120573minus120582119901(119908119909minus120572) (21)

Proof Making use of (7) we get

119863120582

minus[119905120582minus120573

119864120574120575119902

120572120573119901(119908119905minus120572)] (119909)

= (minus1)119898(

119889

119889119909)

1198981

Γ (119898 minus 120582)

times int

infin

119909


119905120582minus120573

infin

sum

119899=0

(120574)119902119899(119908119905minus120572

)119899

(120575)119901119899Γ (120572119899 + 120573)119889119905

= (minus1)119898(

119889

119889119909)

1198981

Γ (119898 minus 120582)

times

infin

sum

119899=0

119908119899(120574)119902119899

(120575)119901119899Γ (120572119899 + 120573)

times int

infin

119909


119905120582minus120573

119905minus120572119899

119889119905

(22)

Let 119906 = (119905 minus 119909)119905 then

119863120582

minus[119905120582minus120573

119864120574120575119902

120572120573119901(119908119905minus120572

)] (119909)

= (minus1)119898(

119889

119889119909)

1198981

Γ (119898 minus 120582)

times

infin

sum

119899=0

(120574)119902119899119908119899119909119898minus120572119899minus120573

(120575)119901119899 Γ (120572119899 + 120573)int

1

0

119906119898minus120582minus1

(1 minus 119906)120572119899+120573minus119898minus1

119889119906

= (minus1)119898(

119889

119889119909)

1198981

Γ (119898 minus 120582)

times

infin

sum

119899=0

(120574)119902119899119908119899119909119898minus120572119899minus120573

(120575)119901119899 Γ (120572119899 + 120573)119861 (119898 minus 120582 120572119899 + 120573 minus 119898)

= (minus1)119898

infin

sum

119899=0

(120574)119902119899119908119899Γ (120572119899 + 120573 minus 119898)

(120575)119901119899Γ (120572119899 + 120573) Γ (120572119899 + 120573 minus 120582)sdot (119898 minus 120582119899 minus 120573)

sdot sdot sdot (119898 minus 120572119899 minus 120573 minus 119898 + 1) 119909119898minus120572119899minus120573minus119898

= (minus1)119898

infin

sum

119899=0

(120574)119902119899119908119899Γ (120572119899 + 120573 minus 119898)

(120575)119901119899Γ (120572119899 + 120573) Γ (120572119899 + 120573 minus 120582)

sdot (minus1)119898(120572119899 + 120573 minus 119898)

119898119909minus120572119899minus120573

= 119909minus120573

infin

sum

119899=0

(120574)119902119899(119908119909minus120572)119899

(120575)119901119899Γ (120572119899 + 120573 minus 120582)

= 119909minus120573

119864120574120575119902

120572120573minus120582119901(119908119909minus120572)

(23)

3 Weyl Integral Operator with GeneralizedMittag-Leffler Function in the Kernel

Consider theWeyl integral operator defined in (9) containing119864120574120575119902

120572120573119901(119911) in the kernel First of all we prove that the operator

E120574120575119902

120572120573119901119908infinis bounded on 119871(119886infin)

Theorem 3 Let 120572 120573 120574 120575 120582 119908 isin C with minRe(120572) Re(120573)Re(120574) Re(120575) Re(120582) gt 0 and 119901 119902 gt 0 then the operatorE120574120575119902

120572120573119901119908infinis bounded on 119871(119886infin) and

100381710038171003817100381710038171003817E120574120575119902

120572120573119901119908infin1205931003817100381710038171003817100381710038171

le 1205731003817100381710038171003817120593

10038171003817100381710038171 (24)

where

120573 = (119887 minus 119886)Re(120573)infin

sum

119899=0

10038161003816100381610038161003816(120574)119902119899

10038161003816100381610038161003816

1003816100381610038161003816119908(119887 minus 119886)1205721003816100381610038161003816

119899

10038161003816100381610038161003816(120575)119901119899

10038161003816100381610038161003816

1003816100381610038161003816Γ (120572119899 + 120573)1003816100381610038161003816

1003816100381610038161003816Re (120572) 119899 + Re (120573)1003816100381610038161003816

(25)

Proof Let 119862119899 denote the 119899th term of (25) then10038161003816100381610038161003816100381610038161003816

119862119899+1

119862119899

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

(120574)119902119899+119902

(120574)119902119899

1003816100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

Γ (120572119899 + 120573)

Γ (120572119899 + 120573 + 120572)

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

(120575)119901119899

(120575)119901119899+119901

100381610038161003816100381610038161003816100381610038161003816

times

100381610038161003816100381610038161003816100381610038161003816

Re (120572) 119899 + Re (120573)Re (120572) 119899 + Re (120572) + Re (120573)

100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816119908(119887 minus 119886)

Re(120572)10038161003816100381610038161003816

asymp

10038161003816100381610038161003816119908(119887 minus 119886)

Re(120572)10038161003816100381610038161003816 (119902119899)119902

(|120572| 119899)Re(120572)

(119901119899)119901

as 119899 997888rarr infin

(26)

Hence |119862119899+1119862119899| rarr 0 as 119899 rarr infin and 119902 lt 119901 + Re(120572) whichmeans that the right-hand side of (25) is convergent and finiteunder the given condition


Now according to (9) (10) and (11) we get100381710038171003817100381710038171003817E120574120575119902

120572120573119901119908infin1205931003817100381710038171003817100381710038171

= int

infin

119886

10038161003816100381610038161003816100381610038161003816

int

infin

119909

(119905 minus 119909)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

10038161003816100381610038161003816100381610038161003816

119889119909

le int

infin

119886

[int

119905

119886

(119905 minus 119909)120573minus1

100381610038161003816100381610038161003816119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572]100381610038161003816100381610038161003816119889119909]

1003816100381610038161003816120593 (119905)1003816100381610038161003816 119889119905

= int

infin

119886

[int

119905minus119886

0

119906Re(120573)minus1 100381610038161003816100381610038161003816

119864120574120575119902

120572120573119901(119908119906120572)100381610038161003816100381610038161003816119889119906]

1003816100381610038161003816120593 (119905)1003816100381610038161003816 119889119905

by letting 119906 = 119905 minus 119909

le int

infin

119886

[int

119887minus119886

0

119906Re(120573)minus1 100381610038161003816100381610038161003816

119864120574120575119902

120572120573119901(119908119906120572)100381610038161003816100381610038161003816119889119906]

1003816100381610038161003816120593 (119905)1003816100381610038161003816 119889119905

(27)

Let

int

119887minus119886

0

119906Re(120573)minus1 100381610038161003816100381610038161003816

119864120574120575119902

120572120573119901(119908119906120572)100381610038161003816100381610038161003816119889119906 = 120573 (28)

then

120573 =

infin

sum

119899=0

10038161003816100381610038161003816(120574)119902119899

10038161003816100381610038161003816

10038161003816100381610038161199081198991003816100381610038161003816

10038161003816100381610038161003816(120575)119901119899

10038161003816100381610038161003816

1003816100381610038161003816Γ (120572119899 + 120573)1003816100381610038161003816

int

119887minus119886

0

119906Re(120572)119899+Re(120573)minus1

119889119906

= (119887 minus 119886)Re(120573)infin

sum

119899=0

10038161003816100381610038161003816(120574)119902119899

10038161003816100381610038161003816

10038161003816100381610038161003816119908(119887 minus 119886)

Re(120572)10038161003816100381610038161003816119899

10038161003816100381610038161003816(120575)119901119899

10038161003816100381610038161003816

1003816100381610038161003816Γ (120572119899 + 120573)1003816100381610038161003816

1003816100381610038161003816Re (120572) 119899 + Re (120573)1003816100381610038161003816

(29)

Hence100381710038171003817100381710038171003817E120574120575119902

120572120573119901119908infin1205931003817100381710038171003817100381710038171

le int

infin

119886

1205731003816100381610038161003816120593 (119905)

1003816100381610038161003816 119889119905 le 1205731003817100381710038171003817120593

10038171003817100381710038171 (30)

We consider now composition ofWeyl fractional integra-tion and differentiation 119868

120582

minus 119863120582

minuswith the operator E120574120575119902

120572120573119901119908infin

defined in (9) contained in the next two theorems

Remark 4 One can use the result of the next lemma for theproof of the stated theorems (see [5])

Lemma 5 Let 119886 isin R+ 120572 120573 120574 120575 120582 isin C minRe(120572) Re(120573)Re(120574) Re(120575) Re(120582) gt 0 and 119901 119902 gt 0 then for 119909 gt 119886 one has

119868120582

119886+[(119905 minus 119886)

120573minus1119864120574120575119902

120572120573119901[119908(119905 minus 119886)

120572]] (119909)

= (119909 minus 119886)120573+120582minus1

119864120574120575119902

120572120573+120582119901[119908(119909 minus 119886)

120572]

(31)

Theorem 6 Let 120572 120573 120574 120575 120582 119908 isin C with minRe(120572) Re(120573)Re(120574) Re(120575) gt 0 and 119901 119902 gt 0 then

(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909) = (E

120574120575119902

120572120573+120582119901119908infin120593) (119909)

= (E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

(32)

Proof Applying (8) and (9) and by using Dirichlet formula(11) yields

(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

=1

Γ (120582)

times (int

infin

119909

(119906 minus 119909)120582minus1

times[int

infin

119906

(119905 minus 119906)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119906)

120572] 120593 (119905) 119889119905]) 119889119906

= int

infin

119909

1

Γ (120582)

times [int

119905

119909

(119906 minus 119909)120582minus1

(119905 minus 119906)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119906)

120572] 119889119906]

times 120593 (119905) 119889119905

(33)Let

120591 = (119905 minus 119906) (34)then

(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= int

infin

119909

1

Γ (120582)[int

119905minus119909

0

(119905 minus 119909 minus 120591)120582minus1

times120591120573minus1

119864120574120575119902

120572120573119901(119908120591120572) 119889119905] 120593 (119905) 119889119905

(35)

Applying (13) and the result of Lemma 5 we get

(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= int

infin

119909

119868120582

0[120591120573minus1

119864120574120575119902

120572120573119901(119908120591120572)] (119905 minus 119909) 120593 (119905) 119889119905

= int

infin

119909

(119905 minus 119909)120573+120582minus1

119864120574120575119902

120572120573+120582119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

= (E120574120575119902

120572120573+120582119901119908infin120593) (119909)

(36)

On the other hand

(E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

= int

infin

119909

(119905 minus 119909)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572]

1

Γ (120582)

times [int

infin

119905

(119906 minus 119905)120582minus1

120593 (119906) 119889119906] 119889119905

= int

infin

119909

1

Γ (120582)[int

119906

119909

(119905 minus 119909)120573minus1

(119906 minus 119905)120582minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572119889119905]]

times 120593 (119906) 119889119906

(37)


Let 120591 = 119905 minus 119909 we get

(E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

= int

infin

119909

1

Γ (120582)[int

119906minus119909

0



119864120574120575119902

120572120573119901(119908120591120572) 119889120591] 120593 (119906) 119889119906

(38)

Returning to (13) and Lemma 5 we have

(E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

= int

infin

119909

119868120582

0[120591120573minus1

119864120574120575119902

120572120573119901(119908120591120572)] (119906 minus 119909) 120593 (119906) 119889119906

= int

infin

119909

(119906 minus 119909)120573+120582minus1

119864120574120575119902

120572120573+120582119901[119908(119906 minus 119909)

120572] 120593 (119906) 119889119906

= (E120574120575119902

120572120573+120582119901119908infin120593) (119909)

(39)

which ends the proof

A similar result concerning the Weyl fractional differen-tiation is stated in the following theorem

Theorem 7 If the condition of Theorem 6 is satisfied then

(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909) = (E

120574120575119902

120572120573minus120582119901119908infin120593) (119909) (40)


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909) = (minus1)

119899(

119889

119889119909)

119899

(119868119899minus120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

(41)

and applyingTheorem 6 yields

(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= (minus1)119899(

119889

119889119909)

119899

times int

infin

119909


119864120574120575119902

120572120573minus119899minus120582119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

(42)

By using Dirichlet formula (12) we get

(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= (minus1)119899(

119889

119889119909)

119899minus1

times(int

infin

119909

120597

120597119909(119905minus119909)

120573+119899minus120582minus1119864120574120575119902

120572120573+119899minus120582119901[119908(119905minus 119909)

120572] 120593 (119905) 119889119905)

+ lim119905rarr119909+


119864120574120575119902

120572120573+119899minus120582119901[119908(119905 minus 119909)

120572] 120593 (119905)

= (minus1)119899(

119889

119889119909)

119899minus1

times int

infin

119909

infin

sum

119899=0

(120574)119902119899119908119899(120572119899 + 120573 + 119899 minus 120582 minus 1)

(120575)119901119899Γ (120572119899 + 120573 minus 119899 minus 120582)

times (119905 minus 119909)120572119899+120573+119899minus120582minus2

sdot 120593 (119905) 119889119905

= (minus1)119899(

119889

119889119909)

119899minus1

int

infin

119909

(minus1) (119905 minus 119909)120573+119899minus120582minus2

times 119864120574120575119902

120572120573+119899minus120582minus1119901[119908(119905 minus 119909)

120572] sdot 120593 (119905) 119889119905

(43)Repeating this process 119899 minus 1 times we get

(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= (minus1)119899(minus1)119899int

infin

119909


119864120574120575119902

120572120573minus120582119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

= (E120574120575119902

120572120573minus120582119901119908infin120593) (119909)

(44)

References

[1] G M Mittag-Leffler ldquoSur la nouvelle function 119864120572(119911)rdquo Comptes

Rendus de lrsquoAcademie des Sciences vol 137 pp 554ndash558 1903[2] A Wiman ldquoUber den Fundamentalsatz in der Teorie der

Funktionen119864120572(119909)rdquoActaMathematica vol 29 no 1 pp 191ndash201

1905[3] T R Prabhakar ldquoA singular integral equationwith a generalized

Mittag-Leffler function in the kernelrdquo Yokohama MathematicalJournal vol 19 pp 7ndash15 1971

[4] A K Shukla and J C Prajapati ldquoOn a generalization of Mittag-Leffler function and its propertiesrdquo Journal of MathematicalAnalysis and Applications vol 336 no 2 pp 797ndash811 2007

[5] T O Salim and A W Faraj ldquoA generalization of Mittag-Leffler function and integral operator associated with fractionalcalculusrdquo Journal of Fractional Calculus and Applications vol 3no 5 pp 1ndash13 2012

[6] A A Kilbas M Saigo and R K Saxena ldquoGeneralized Mittag-Leffler function and generalized fractional calculus operatorsrdquoIntegral Transforms and Special Functions vol 15 no 1 pp 31ndash49 2004

[7] H M Srivastava and Z Tomovski ldquoFractional calculus with anintegral operator containing a generalized Mittag-Leffler func-tion in the kernelrdquo Applied Mathematics and Computation vol211 no 1 pp 198ndash210 2009

[8] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Publishers Yverdon Switzerland 1993

[9] IN SneddonTheUse of Integral Transforms TataMcGrawHillNew Delhi India 1979

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


On the other hand Salim and Faraj in their research paperdefined and studied an integral operator E120574120575119902

120572120573119901119908119886+as

(E120574120575119902

120572120573119901119908119886+120593) (119909) = int

119909

119886

(119909 minus 119905)120573minus1

119864120574120575119902

120572120573119901[119908(119909 minus 119905)

120572] 120593 (119905) 119889119905

(5)

containing 119864120574120575119902

120572120573119901(119911) in the kernel Also composition of

Riemann-Liouville fractional integral and differential opera-tors with the integral operator defined in (5) was established

This paper is devoted for the study of further propertiesof the generalizedMittag-Leffler function119864

120574120575119902

120572120573119901(119911) defined in

(3) with another type of fractional calculus operators calledWeyl fractional integral and differential operators written as

(119868120582

minus120593) (119909) =

1

Γ (120582)int

infin

119909

(119905 minus 119909)120582minus1

120593 (119905) 119889119905 (6)

(119863120582

minus120593) (119909)

= (minus1)119898(

119889

119889119909)

1198981

Γ (119898 minus 120582)int

infin

119909


120593 (119905) 119889119905

(7)

The last definition can be written in the form

(119863120582

minus120593) (119909) = (minus1)

119898(

119889

119889119909)

119898

(119868119898minus120582

minus120593) (119909) (8)

Precisely the authors investigate the basic properties of Weylfractional integral and differential operator with generalizedMittag-Leffler function 119864

120574120575119902

120572120573119901(119911) moreover a new integral

operator depending onWeyl fractional integral operator andcontaining 119864

120574120575119902

120572120573119901(119911) in its kernel is established as

(E120574120575119902

120572120573119901119908infin120593) (119908) = int

infin

119909

(119905 minus 119909)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

(9)

The condition of boundedness of the integral operator (9)is discussed and stated in the space 119871(119886infin) of Lebesgue-measurable functions on (119886infin)

119871 (119886infin) = 119892 (119909) 1003817100381710038171003817119892

10038171003817100381710038171= int

infin

119886

1003816100381610038161003816119892 (119909)1003816100381610038161003816 119889119909 lt infin (10)

Also composition of Weyl fractional integration and differ-entiation with the operator defined in (9) is established

Throughout this paper we need the followingwell-knownfacts and rules

(i) Fubinirsquos theorem (Dirichlet formula) [8]

int

119887

119886

119889119909int

119909

119886

119891 (119909 119905) 119889119905 = int

119887

119886

119889119905int

119887

119905

119891 (119909 119905) 119889119909 (11)

119889

119889119909int

119909

119886

ℎ (119909 119905) 119889119905 = [int

119909

119886

120597

120597119909ℎ (119909 119905) 119889119905] + ℎ (119909 119909) (12)

(ii) The Riemann-Liouville fractional integral [8]

(119868120582

119886+120593) (119909) =

1

Γ (120582)int

119909

119886

(119909 minus 119905)120582minus1

120593 (119905) 119889119905

(120572 isin CRe (120572) gt 0)

(13)

(iii) The Riemann-Liouville fractional derivative [8]

(119863120582

119886+120593) (119909) = (

119889

119889119909)

119899

(119868119899minus120582

119886+120593) (119909) 119899 = [Re (120572)] + 1 (14)

(iv) Beta transform (Sneddon [9])

119861 119891 (119911) 119886 119887 = int

1

0

119911119886minus1

(1 minus 119911)119887119891 (119911) 119889119911 (15)

where Re(119886) gt 0 Re(119887) gt 0

(v) The Beta function is written as

120573 (120572 120573) =Γ (120572) Γ (120573)

Γ (120572 + 120573) (16)

(vi) The difference property of the Gamma function is

Γ (120572 + 1) = 120572Γ (120572) (17)

2 Further Properties of Weyl FractionalIntegral Related to Mittag-Leffler Function

In this section we consider composition of Weyl fractionalintegral and derivative (6) and (7) with generalized Mittag-Leffler function 119864

120574120575119902

120572120573119901(119911) defined in (3)

Theorem 1 Let 120572 120573 120574 120575 120582 119908 isin CminRe(120572) Re(120573) Re(120574)Re(120575) Re(120582) gt 0 and 119901 119902 gt 0 then

119868120582


119864120574120575119902

120572120573119901(119908119905minus120572)] (119909) = 119909

minus120573119864120574120575119902

120572120573+120582119901(119908119909minus120572) (18)

Proof

119868120582


119864120574120575119902

120572120573119901(119908119905minus120572)] (119909)

=1

Γ (120582)int

infin

119909

(119905 minus 119909)120582minus1

119905minus120582minus120573

times

infin

sum

119899=0

(120574)119902119899119908119899119905minus120572119899

(120575)119901119899Γ (120572119899 + 120573)119889119905

=1

Γ (120582)

infin

sum

119899=0

(120574)119902119899119908119899

(120575)119901119899Γ (120572119899 + 120573)

times int

infin

119909

(119905 minus 119909)120582minus1

119905minus120582minus120573

119905minus120572119889119905

(19)


Let 119906 = (119905 minus 119909)119905 then

119868120582


119864120574120575119902

120572120573119901(119908119905minus120572

)] (119909)

=1

Γ (120582)

infin

sum

119899=0

(120574)119902119899119908119899119909minus120573minus120572119899

(120575)119901119899Γ (120572119899 + 120573)

times int

1

0

119906120582minus1

(1 minus 119906)120572119899+120573minus1

119889119906

=1

Γ (120582)

infin

sum

119899=0

(120574)119902119899119908119899119909minus120573minus120572119899

(120575)119901119899Γ (120572119899 + 120573)119861 (120582 120572119899 + 120573)

= 119909minus120573

119864120574120575119902

120572120573+120582119901(119908119909minus120572)

(20)


119863120582

minus[119905120582minus120573

119864120574120575119902

120572120573119901(119908119905minus120572)] (119909) = 119909

minus120573119864120574120575

120572120573minus120582119901(119908119909minus120572) (21)


119863120582

minus[119905120582minus120573

119864120574120575119902

120572120573119901(119908119905minus120572)] (119909)

= (minus1)119898(

119889

119889119909)

1198981

Γ (119898 minus 120582)

times int

infin

119909


119905120582minus120573

infin

sum

119899=0

(120574)119902119899(119908119905minus120572

)119899

(120575)119901119899Γ (120572119899 + 120573)119889119905

= (minus1)119898(

119889

119889119909)

1198981

Γ (119898 minus 120582)

times

infin

sum

119899=0

119908119899(120574)119902119899

(120575)119901119899Γ (120572119899 + 120573)

times int

infin

119909


119905120582minus120573

119905minus120572119899

119889119905

(22)

Let 119906 = (119905 minus 119909)119905 then

119863120582

minus[119905120582minus120573

119864120574120575119902

120572120573119901(119908119905minus120572

)] (119909)

= (minus1)119898(

119889

119889119909)

1198981

Γ (119898 minus 120582)

times

infin

sum

119899=0

(120574)119902119899119908119899119909119898minus120572119899minus120573

(120575)119901119899 Γ (120572119899 + 120573)int

1

0

119906119898minus120582minus1


119889119906

= (minus1)119898(

119889

119889119909)

1198981

Γ (119898 minus 120582)

times

infin

sum

119899=0

(120574)119902119899119908119899119909119898minus120572119899minus120573

(120575)119901119899 Γ (120572119899 + 120573)119861 (119898 minus 120582 120572119899 + 120573 minus 119898)

= (minus1)119898

infin

sum

119899=0

(120574)119902119899119908119899Γ (120572119899 + 120573 minus 119898)



= (minus1)119898

infin

sum

119899=0

(120574)119902119899119908119899Γ (120572119899 + 120573 minus 119898)

(120575)119901119899Γ (120572119899 + 120573) Γ (120572119899 + 120573 minus 120582)

sdot (minus1)119898(120572119899 + 120573 minus 119898)

119898119909minus120572119899minus120573

= 119909minus120573

infin

sum

119899=0

(120574)119902119899(119908119909minus120572)119899

(120575)119901119899Γ (120572119899 + 120573 minus 120582)

= 119909minus120573

119864120574120575119902

120572120573minus120582119901(119908119909minus120572)

(23)




E120574120575119902




100381710038171003817100381710038171003817E120574120575119902

120572120573119901119908infin1205931003817100381710038171003817100381710038171

le 1205731003817100381710038171003817120593

10038171003817100381710038171 (24)

where

120573 = (119887 minus 119886)Re(120573)infin

sum

119899=0

10038161003816100381610038161003816(120574)119902119899

10038161003816100381610038161003816

1003816100381610038161003816119908(119887 minus 119886)1205721003816100381610038161003816

119899

10038161003816100381610038161003816(120575)119901119899

10038161003816100381610038161003816

1003816100381610038161003816Γ (120572119899 + 120573)1003816100381610038161003816

1003816100381610038161003816Re (120572) 119899 + Re (120573)1003816100381610038161003816

(25)


119862119899+1

119862119899

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

(120574)119902119899+119902

(120574)119902119899

1003816100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

Γ (120572119899 + 120573)

Γ (120572119899 + 120573 + 120572)

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

(120575)119901119899

(120575)119901119899+119901

100381610038161003816100381610038161003816100381610038161003816

times

100381610038161003816100381610038161003816100381610038161003816

Re (120572) 119899 + Re (120573)Re (120572) 119899 + Re (120572) + Re (120573)

100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816119908(119887 minus 119886)

Re(120572)10038161003816100381610038161003816

asymp

10038161003816100381610038161003816119908(119887 minus 119886)

Re(120572)10038161003816100381610038161003816 (119902119899)119902

(|120572| 119899)Re(120572)

(119901119899)119901


(26)




120572120573119901119908infin1205931003817100381710038171003817100381710038171

= int

infin

119886

10038161003816100381610038161003816100381610038161003816

int

infin

119909

(119905 minus 119909)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

10038161003816100381610038161003816100381610038161003816

119889119909

le int

infin

119886

[int

119905

119886

(119905 minus 119909)120573minus1

100381610038161003816100381610038161003816119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572]100381610038161003816100381610038161003816119889119909]

1003816100381610038161003816120593 (119905)1003816100381610038161003816 119889119905

= int

infin

119886

[int

119905minus119886

0

119906Re(120573)minus1 100381610038161003816100381610038161003816

119864120574120575119902

120572120573119901(119908119906120572)100381610038161003816100381610038161003816119889119906]

1003816100381610038161003816120593 (119905)1003816100381610038161003816 119889119905

by letting 119906 = 119905 minus 119909

le int

infin

119886

[int

119887minus119886

0

119906Re(120573)minus1 100381610038161003816100381610038161003816

119864120574120575119902

120572120573119901(119908119906120572)100381610038161003816100381610038161003816119889119906]

1003816100381610038161003816120593 (119905)1003816100381610038161003816 119889119905

(27)

Let

int

119887minus119886

0

119906Re(120573)minus1 100381610038161003816100381610038161003816

119864120574120575119902

120572120573119901(119908119906120572)100381610038161003816100381610038161003816119889119906 = 120573 (28)

then

120573 =

infin

sum

119899=0

10038161003816100381610038161003816(120574)119902119899

10038161003816100381610038161003816

10038161003816100381610038161199081198991003816100381610038161003816

10038161003816100381610038161003816(120575)119901119899

10038161003816100381610038161003816

1003816100381610038161003816Γ (120572119899 + 120573)1003816100381610038161003816

int

119887minus119886

0

119906Re(120572)119899+Re(120573)minus1

119889119906

= (119887 minus 119886)Re(120573)infin

sum

119899=0

10038161003816100381610038161003816(120574)119902119899

10038161003816100381610038161003816

10038161003816100381610038161003816119908(119887 minus 119886)

Re(120572)10038161003816100381610038161003816119899

10038161003816100381610038161003816(120575)119901119899

10038161003816100381610038161003816

1003816100381610038161003816Γ (120572119899 + 120573)1003816100381610038161003816

1003816100381610038161003816Re (120572) 119899 + Re (120573)1003816100381610038161003816

(29)

Hence100381710038171003817100381710038171003817E120574120575119902

120572120573119901119908infin1205931003817100381710038171003817100381710038171

le int

infin

119886

1205731003816100381610038161003816120593 (119905)

1003816100381610038161003816 119889119905 le 1205731003817100381710038171003817120593

10038171003817100381710038171 (30)


120582

minus 119863120582


120572120573119901119908infin




119868120582

119886+[(119905 minus 119886)

120573minus1119864120574120575119902

120572120573119901[119908(119905 minus 119886)

120572]] (119909)

= (119909 minus 119886)120573+120582minus1

119864120574120575119902

120572120573+120582119901[119908(119909 minus 119886)

120572]

(31)


(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909) = (E

120574120575119902

120572120573+120582119901119908infin120593) (119909)

= (E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

(32)


(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

=1

Γ (120582)

times (int

infin

119909

(119906 minus 119909)120582minus1

times[int

infin

119906

(119905 minus 119906)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119906)

120572] 120593 (119905) 119889119905]) 119889119906

= int

infin

119909

1

Γ (120582)

times [int

119905

119909

(119906 minus 119909)120582minus1

(119905 minus 119906)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119906)

120572] 119889119906]

times 120593 (119905) 119889119905

(33)Let

120591 = (119905 minus 119906) (34)then

(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= int

infin

119909

1

Γ (120582)[int

119905minus119909

0



119864120574120575119902

120572120573119901(119908120591120572) 119889119905] 120593 (119905) 119889119905

(35)


(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= int

infin

119909

119868120582

0[120591120573minus1

119864120574120575119902

120572120573119901(119908120591120572)] (119905 minus 119909) 120593 (119905) 119889119905

= int

infin

119909

(119905 minus 119909)120573+120582minus1

119864120574120575119902

120572120573+120582119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

= (E120574120575119902

120572120573+120582119901119908infin120593) (119909)

(36)

On the other hand

(E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

= int

infin

119909

(119905 minus 119909)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572]

1

Γ (120582)

times [int

infin

119905

(119906 minus 119905)120582minus1

120593 (119906) 119889119906] 119889119905

= int

infin

119909

1

Γ (120582)[int

119906

119909

(119905 minus 119909)120573minus1

(119906 minus 119905)120582minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572119889119905]]

times 120593 (119906) 119889119906

(37)


Let 120591 = 119905 minus 119909 we get

(E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

= int

infin

119909

1

Γ (120582)[int

119906minus119909

0



119864120574120575119902

120572120573119901(119908120591120572) 119889120591] 120593 (119906) 119889119906

(38)


(E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

= int

infin

119909

119868120582

0[120591120573minus1

119864120574120575119902

120572120573119901(119908120591120572)] (119906 minus 119909) 120593 (119906) 119889119906

= int

infin

119909

(119906 minus 119909)120573+120582minus1

119864120574120575119902

120572120573+120582119901[119908(119906 minus 119909)

120572] 120593 (119906) 119889119906

= (E120574120575119902

120572120573+120582119901119908infin120593) (119909)

(39)




(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909) = (E

120574120575119902

120572120573minus120582119901119908infin120593) (119909) (40)


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909) = (minus1)

119899(

119889

119889119909)

119899

(119868119899minus120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

(41)


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= (minus1)119899(

119889

119889119909)

119899

times int

infin

119909


119864120574120575119902


120572] 120593 (119905) 119889119905

(42)


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= (minus1)119899(

119889

119889119909)

119899minus1

times(int

infin

119909

120597

120597119909(119905minus119909)

120573+119899minus120582minus1119864120574120575119902

120572120573+119899minus120582119901[119908(119905minus 119909)

120572] 120593 (119905) 119889119905)

+ lim119905rarr119909+


119864120574120575119902

120572120573+119899minus120582119901[119908(119905 minus 119909)

120572] 120593 (119905)

= (minus1)119899(

119889

119889119909)

119899minus1

times int

infin

119909

infin

sum

119899=0

(120574)119902119899119908119899(120572119899 + 120573 + 119899 minus 120582 minus 1)

(120575)119901119899Γ (120572119899 + 120573 minus 119899 minus 120582)


sdot 120593 (119905) 119889119905

= (minus1)119899(

119889

119889119909)

119899minus1

int

infin

119909


times 119864120574120575119902


120572] sdot 120593 (119905) 119889119905


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= (minus1)119899(minus1)119899int

infin

119909


119864120574120575119902

120572120573minus120582119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

= (E120574120575119902

120572120573minus120582119901119908infin120593) (119909)

(44)

References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Let 119906 = (119905 minus 119909)119905 then

119868120582


119864120574120575119902

120572120573119901(119908119905minus120572

)] (119909)

=1

Γ (120582)

infin

sum

119899=0

(120574)119902119899119908119899119909minus120573minus120572119899

(120575)119901119899Γ (120572119899 + 120573)

times int

1

0

119906120582minus1

(1 minus 119906)120572119899+120573minus1

119889119906

=1

Γ (120582)

infin

sum

119899=0

(120574)119902119899119908119899119909minus120573minus120572119899

(120575)119901119899Γ (120572119899 + 120573)119861 (120582 120572119899 + 120573)

= 119909minus120573

119864120574120575119902

120572120573+120582119901(119908119909minus120572)

(20)


119863120582

minus[119905120582minus120573

119864120574120575119902

120572120573119901(119908119905minus120572)] (119909) = 119909

minus120573119864120574120575

120572120573minus120582119901(119908119909minus120572) (21)


119863120582

minus[119905120582minus120573

119864120574120575119902

120572120573119901(119908119905minus120572)] (119909)

= (minus1)119898(

119889

119889119909)

1198981

Γ (119898 minus 120582)

times int

infin

119909


119905120582minus120573

infin

sum

119899=0

(120574)119902119899(119908119905minus120572

)119899

(120575)119901119899Γ (120572119899 + 120573)119889119905

= (minus1)119898(

119889

119889119909)

1198981

Γ (119898 minus 120582)

times

infin

sum

119899=0

119908119899(120574)119902119899

(120575)119901119899Γ (120572119899 + 120573)

times int

infin

119909


119905120582minus120573

119905minus120572119899

119889119905

(22)

Let 119906 = (119905 minus 119909)119905 then

119863120582

minus[119905120582minus120573

119864120574120575119902

120572120573119901(119908119905minus120572

)] (119909)

= (minus1)119898(

119889

119889119909)

1198981

Γ (119898 minus 120582)

times

infin

sum

119899=0

(120574)119902119899119908119899119909119898minus120572119899minus120573

(120575)119901119899 Γ (120572119899 + 120573)int

1

0

119906119898minus120582minus1


119889119906

= (minus1)119898(

119889

119889119909)

1198981

Γ (119898 minus 120582)

times

infin

sum

119899=0

(120574)119902119899119908119899119909119898minus120572119899minus120573

(120575)119901119899 Γ (120572119899 + 120573)119861 (119898 minus 120582 120572119899 + 120573 minus 119898)

= (minus1)119898

infin

sum

119899=0

(120574)119902119899119908119899Γ (120572119899 + 120573 minus 119898)



= (minus1)119898

infin

sum

119899=0

(120574)119902119899119908119899Γ (120572119899 + 120573 minus 119898)

(120575)119901119899Γ (120572119899 + 120573) Γ (120572119899 + 120573 minus 120582)

sdot (minus1)119898(120572119899 + 120573 minus 119898)

119898119909minus120572119899minus120573

= 119909minus120573

infin

sum

119899=0

(120574)119902119899(119908119909minus120572)119899

(120575)119901119899Γ (120572119899 + 120573 minus 120582)

= 119909minus120573

119864120574120575119902

120572120573minus120582119901(119908119909minus120572)

(23)




E120574120575119902




100381710038171003817100381710038171003817E120574120575119902

120572120573119901119908infin1205931003817100381710038171003817100381710038171

le 1205731003817100381710038171003817120593

10038171003817100381710038171 (24)

where

120573 = (119887 minus 119886)Re(120573)infin

sum

119899=0

10038161003816100381610038161003816(120574)119902119899

10038161003816100381610038161003816

1003816100381610038161003816119908(119887 minus 119886)1205721003816100381610038161003816

119899

10038161003816100381610038161003816(120575)119901119899

10038161003816100381610038161003816

1003816100381610038161003816Γ (120572119899 + 120573)1003816100381610038161003816

1003816100381610038161003816Re (120572) 119899 + Re (120573)1003816100381610038161003816

(25)


119862119899+1

119862119899

10038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

(120574)119902119899+119902

(120574)119902119899

1003816100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

Γ (120572119899 + 120573)

Γ (120572119899 + 120573 + 120572)

100381610038161003816100381610038161003816100381610038161003816

100381610038161003816100381610038161003816100381610038161003816

(120575)119901119899

(120575)119901119899+119901

100381610038161003816100381610038161003816100381610038161003816

times

100381610038161003816100381610038161003816100381610038161003816

Re (120572) 119899 + Re (120573)Re (120572) 119899 + Re (120572) + Re (120573)

100381610038161003816100381610038161003816100381610038161003816

10038161003816100381610038161003816119908(119887 minus 119886)

Re(120572)10038161003816100381610038161003816

asymp

10038161003816100381610038161003816119908(119887 minus 119886)

Re(120572)10038161003816100381610038161003816 (119902119899)119902

(|120572| 119899)Re(120572)

(119901119899)119901


(26)




120572120573119901119908infin1205931003817100381710038171003817100381710038171

= int

infin

119886

10038161003816100381610038161003816100381610038161003816

int

infin

119909

(119905 minus 119909)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

10038161003816100381610038161003816100381610038161003816

119889119909

le int

infin

119886

[int

119905

119886

(119905 minus 119909)120573minus1

100381610038161003816100381610038161003816119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572]100381610038161003816100381610038161003816119889119909]

1003816100381610038161003816120593 (119905)1003816100381610038161003816 119889119905

= int

infin

119886

[int

119905minus119886

0

119906Re(120573)minus1 100381610038161003816100381610038161003816

119864120574120575119902

120572120573119901(119908119906120572)100381610038161003816100381610038161003816119889119906]

1003816100381610038161003816120593 (119905)1003816100381610038161003816 119889119905

by letting 119906 = 119905 minus 119909

le int

infin

119886

[int

119887minus119886

0

119906Re(120573)minus1 100381610038161003816100381610038161003816

119864120574120575119902

120572120573119901(119908119906120572)100381610038161003816100381610038161003816119889119906]

1003816100381610038161003816120593 (119905)1003816100381610038161003816 119889119905

(27)

Let

int

119887minus119886

0

119906Re(120573)minus1 100381610038161003816100381610038161003816

119864120574120575119902

120572120573119901(119908119906120572)100381610038161003816100381610038161003816119889119906 = 120573 (28)

then

120573 =

infin

sum

119899=0

10038161003816100381610038161003816(120574)119902119899

10038161003816100381610038161003816

10038161003816100381610038161199081198991003816100381610038161003816

10038161003816100381610038161003816(120575)119901119899

10038161003816100381610038161003816

1003816100381610038161003816Γ (120572119899 + 120573)1003816100381610038161003816

int

119887minus119886

0

119906Re(120572)119899+Re(120573)minus1

119889119906

= (119887 minus 119886)Re(120573)infin

sum

119899=0

10038161003816100381610038161003816(120574)119902119899

10038161003816100381610038161003816

10038161003816100381610038161003816119908(119887 minus 119886)

Re(120572)10038161003816100381610038161003816119899

10038161003816100381610038161003816(120575)119901119899

10038161003816100381610038161003816

1003816100381610038161003816Γ (120572119899 + 120573)1003816100381610038161003816

1003816100381610038161003816Re (120572) 119899 + Re (120573)1003816100381610038161003816

(29)

Hence100381710038171003817100381710038171003817E120574120575119902

120572120573119901119908infin1205931003817100381710038171003817100381710038171

le int

infin

119886

1205731003816100381610038161003816120593 (119905)

1003816100381610038161003816 119889119905 le 1205731003817100381710038171003817120593

10038171003817100381710038171 (30)


120582

minus 119863120582


120572120573119901119908infin




119868120582

119886+[(119905 minus 119886)

120573minus1119864120574120575119902

120572120573119901[119908(119905 minus 119886)

120572]] (119909)

= (119909 minus 119886)120573+120582minus1

119864120574120575119902

120572120573+120582119901[119908(119909 minus 119886)

120572]

(31)


(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909) = (E

120574120575119902

120572120573+120582119901119908infin120593) (119909)

= (E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

(32)


(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

=1

Γ (120582)

times (int

infin

119909

(119906 minus 119909)120582minus1

times[int

infin

119906

(119905 minus 119906)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119906)

120572] 120593 (119905) 119889119905]) 119889119906

= int

infin

119909

1

Γ (120582)

times [int

119905

119909

(119906 minus 119909)120582minus1

(119905 minus 119906)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119906)

120572] 119889119906]

times 120593 (119905) 119889119905

(33)Let

120591 = (119905 minus 119906) (34)then

(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= int

infin

119909

1

Γ (120582)[int

119905minus119909

0



119864120574120575119902

120572120573119901(119908120591120572) 119889119905] 120593 (119905) 119889119905

(35)


(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= int

infin

119909

119868120582

0[120591120573minus1

119864120574120575119902

120572120573119901(119908120591120572)] (119905 minus 119909) 120593 (119905) 119889119905

= int

infin

119909

(119905 minus 119909)120573+120582minus1

119864120574120575119902

120572120573+120582119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

= (E120574120575119902

120572120573+120582119901119908infin120593) (119909)

(36)

On the other hand

(E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

= int

infin

119909

(119905 minus 119909)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572]

1

Γ (120582)

times [int

infin

119905

(119906 minus 119905)120582minus1

120593 (119906) 119889119906] 119889119905

= int

infin

119909

1

Γ (120582)[int

119906

119909

(119905 minus 119909)120573minus1

(119906 minus 119905)120582minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572119889119905]]

times 120593 (119906) 119889119906

(37)


Let 120591 = 119905 minus 119909 we get

(E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

= int

infin

119909

1

Γ (120582)[int

119906minus119909

0



119864120574120575119902

120572120573119901(119908120591120572) 119889120591] 120593 (119906) 119889119906

(38)


(E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

= int

infin

119909

119868120582

0[120591120573minus1

119864120574120575119902

120572120573119901(119908120591120572)] (119906 minus 119909) 120593 (119906) 119889119906

= int

infin

119909

(119906 minus 119909)120573+120582minus1

119864120574120575119902

120572120573+120582119901[119908(119906 minus 119909)

120572] 120593 (119906) 119889119906

= (E120574120575119902

120572120573+120582119901119908infin120593) (119909)

(39)




(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909) = (E

120574120575119902

120572120573minus120582119901119908infin120593) (119909) (40)


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909) = (minus1)

119899(

119889

119889119909)

119899

(119868119899minus120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

(41)


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= (minus1)119899(

119889

119889119909)

119899

times int

infin

119909


119864120574120575119902


120572] 120593 (119905) 119889119905

(42)


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= (minus1)119899(

119889

119889119909)

119899minus1

times(int

infin

119909

120597

120597119909(119905minus119909)

120573+119899minus120582minus1119864120574120575119902

120572120573+119899minus120582119901[119908(119905minus 119909)

120572] 120593 (119905) 119889119905)

+ lim119905rarr119909+


119864120574120575119902

120572120573+119899minus120582119901[119908(119905 minus 119909)

120572] 120593 (119905)

= (minus1)119899(

119889

119889119909)

119899minus1

times int

infin

119909

infin

sum

119899=0

(120574)119902119899119908119899(120572119899 + 120573 + 119899 minus 120582 minus 1)

(120575)119901119899Γ (120572119899 + 120573 minus 119899 minus 120582)


sdot 120593 (119905) 119889119905

= (minus1)119899(

119889

119889119909)

119899minus1

int

infin

119909


times 119864120574120575119902


120572] sdot 120593 (119905) 119889119905


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= (minus1)119899(minus1)119899int

infin

119909


119864120574120575119902

120572120573minus120582119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

= (E120574120575119902

120572120573minus120582119901119908infin120593) (119909)

(44)

References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120572120573119901119908infin1205931003817100381710038171003817100381710038171

= int

infin

119886

10038161003816100381610038161003816100381610038161003816

int

infin

119909

(119905 minus 119909)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

10038161003816100381610038161003816100381610038161003816

119889119909

le int

infin

119886

[int

119905

119886

(119905 minus 119909)120573minus1

100381610038161003816100381610038161003816119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572]100381610038161003816100381610038161003816119889119909]

1003816100381610038161003816120593 (119905)1003816100381610038161003816 119889119905

= int

infin

119886

[int

119905minus119886

0

119906Re(120573)minus1 100381610038161003816100381610038161003816

119864120574120575119902

120572120573119901(119908119906120572)100381610038161003816100381610038161003816119889119906]

1003816100381610038161003816120593 (119905)1003816100381610038161003816 119889119905

by letting 119906 = 119905 minus 119909

le int

infin

119886

[int

119887minus119886

0

119906Re(120573)minus1 100381610038161003816100381610038161003816

119864120574120575119902

120572120573119901(119908119906120572)100381610038161003816100381610038161003816119889119906]

1003816100381610038161003816120593 (119905)1003816100381610038161003816 119889119905

(27)

Let

int

119887minus119886

0

119906Re(120573)minus1 100381610038161003816100381610038161003816

119864120574120575119902

120572120573119901(119908119906120572)100381610038161003816100381610038161003816119889119906 = 120573 (28)

then

120573 =

infin

sum

119899=0

10038161003816100381610038161003816(120574)119902119899

10038161003816100381610038161003816

10038161003816100381610038161199081198991003816100381610038161003816

10038161003816100381610038161003816(120575)119901119899

10038161003816100381610038161003816

1003816100381610038161003816Γ (120572119899 + 120573)1003816100381610038161003816

int

119887minus119886

0

119906Re(120572)119899+Re(120573)minus1

119889119906

= (119887 minus 119886)Re(120573)infin

sum

119899=0

10038161003816100381610038161003816(120574)119902119899

10038161003816100381610038161003816

10038161003816100381610038161003816119908(119887 minus 119886)

Re(120572)10038161003816100381610038161003816119899

10038161003816100381610038161003816(120575)119901119899

10038161003816100381610038161003816

1003816100381610038161003816Γ (120572119899 + 120573)1003816100381610038161003816

1003816100381610038161003816Re (120572) 119899 + Re (120573)1003816100381610038161003816

(29)

Hence100381710038171003817100381710038171003817E120574120575119902

120572120573119901119908infin1205931003817100381710038171003817100381710038171

le int

infin

119886

1205731003816100381610038161003816120593 (119905)

1003816100381610038161003816 119889119905 le 1205731003817100381710038171003817120593

10038171003817100381710038171 (30)


120582

minus 119863120582


120572120573119901119908infin




119868120582

119886+[(119905 minus 119886)

120573minus1119864120574120575119902

120572120573119901[119908(119905 minus 119886)

120572]] (119909)

= (119909 minus 119886)120573+120582minus1

119864120574120575119902

120572120573+120582119901[119908(119909 minus 119886)

120572]

(31)


(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909) = (E

120574120575119902

120572120573+120582119901119908infin120593) (119909)

= (E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

(32)


(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

=1

Γ (120582)

times (int

infin

119909

(119906 minus 119909)120582minus1

times[int

infin

119906

(119905 minus 119906)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119906)

120572] 120593 (119905) 119889119905]) 119889119906

= int

infin

119909

1

Γ (120582)

times [int

119905

119909

(119906 minus 119909)120582minus1

(119905 minus 119906)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119906)

120572] 119889119906]

times 120593 (119905) 119889119905

(33)Let

120591 = (119905 minus 119906) (34)then

(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= int

infin

119909

1

Γ (120582)[int

119905minus119909

0



119864120574120575119902

120572120573119901(119908120591120572) 119889119905] 120593 (119905) 119889119905

(35)


(119868120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= int

infin

119909

119868120582

0[120591120573minus1

119864120574120575119902

120572120573119901(119908120591120572)] (119905 minus 119909) 120593 (119905) 119889119905

= int

infin

119909

(119905 minus 119909)120573+120582minus1

119864120574120575119902

120572120573+120582119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

= (E120574120575119902

120572120573+120582119901119908infin120593) (119909)

(36)

On the other hand

(E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

= int

infin

119909

(119905 minus 119909)120573minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572]

1

Γ (120582)

times [int

infin

119905

(119906 minus 119905)120582minus1

120593 (119906) 119889119906] 119889119905

= int

infin

119909

1

Γ (120582)[int

119906

119909

(119905 minus 119909)120573minus1

(119906 minus 119905)120582minus1

119864120574120575119902

120572120573119901[119908(119905 minus 119909)

120572119889119905]]

times 120593 (119906) 119889119906

(37)


Let 120591 = 119905 minus 119909 we get

(E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

= int

infin

119909

1

Γ (120582)[int

119906minus119909

0



119864120574120575119902

120572120573119901(119908120591120572) 119889120591] 120593 (119906) 119889119906

(38)


(E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

= int

infin

119909

119868120582

0[120591120573minus1

119864120574120575119902

120572120573119901(119908120591120572)] (119906 minus 119909) 120593 (119906) 119889119906

= int

infin

119909

(119906 minus 119909)120573+120582minus1

119864120574120575119902

120572120573+120582119901[119908(119906 minus 119909)

120572] 120593 (119906) 119889119906

= (E120574120575119902

120572120573+120582119901119908infin120593) (119909)

(39)




(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909) = (E

120574120575119902

120572120573minus120582119901119908infin120593) (119909) (40)


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909) = (minus1)

119899(

119889

119889119909)

119899

(119868119899minus120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

(41)


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= (minus1)119899(

119889

119889119909)

119899

times int

infin

119909


119864120574120575119902


120572] 120593 (119905) 119889119905

(42)


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= (minus1)119899(

119889

119889119909)

119899minus1

times(int

infin

119909

120597

120597119909(119905minus119909)

120573+119899minus120582minus1119864120574120575119902

120572120573+119899minus120582119901[119908(119905minus 119909)

120572] 120593 (119905) 119889119905)

+ lim119905rarr119909+


119864120574120575119902

120572120573+119899minus120582119901[119908(119905 minus 119909)

120572] 120593 (119905)

= (minus1)119899(

119889

119889119909)

119899minus1

times int

infin

119909

infin

sum

119899=0

(120574)119902119899119908119899(120572119899 + 120573 + 119899 minus 120582 minus 1)

(120575)119901119899Γ (120572119899 + 120573 minus 119899 minus 120582)


sdot 120593 (119905) 119889119905

= (minus1)119899(

119889

119889119909)

119899minus1

int

infin

119909


times 119864120574120575119902


120572] sdot 120593 (119905) 119889119905


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= (minus1)119899(minus1)119899int

infin

119909


119864120574120575119902

120572120573minus120582119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

= (E120574120575119902

120572120573minus120582119901119908infin120593) (119909)

(44)

References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Let 120591 = 119905 minus 119909 we get

(E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

= int

infin

119909

1

Γ (120582)[int

119906minus119909

0



119864120574120575119902

120572120573119901(119908120591120572) 119889120591] 120593 (119906) 119889119906

(38)


(E120574120575119902

120572120573119901119908infin119868120582

minus120593) (119909)

= int

infin

119909

119868120582

0[120591120573minus1

119864120574120575119902

120572120573119901(119908120591120572)] (119906 minus 119909) 120593 (119906) 119889119906

= int

infin

119909

(119906 minus 119909)120573+120582minus1

119864120574120575119902

120572120573+120582119901[119908(119906 minus 119909)

120572] 120593 (119906) 119889119906

= (E120574120575119902

120572120573+120582119901119908infin120593) (119909)

(39)




(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909) = (E

120574120575119902

120572120573minus120582119901119908infin120593) (119909) (40)


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909) = (minus1)

119899(

119889

119889119909)

119899

(119868119899minus120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

(41)


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= (minus1)119899(

119889

119889119909)

119899

times int

infin

119909


119864120574120575119902


120572] 120593 (119905) 119889119905

(42)


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= (minus1)119899(

119889

119889119909)

119899minus1

times(int

infin

119909

120597

120597119909(119905minus119909)

120573+119899minus120582minus1119864120574120575119902

120572120573+119899minus120582119901[119908(119905minus 119909)

120572] 120593 (119905) 119889119905)

+ lim119905rarr119909+


119864120574120575119902

120572120573+119899minus120582119901[119908(119905 minus 119909)

120572] 120593 (119905)

= (minus1)119899(

119889

119889119909)

119899minus1

times int

infin

119909

infin

sum

119899=0

(120574)119902119899119908119899(120572119899 + 120573 + 119899 minus 120582 minus 1)

(120575)119901119899Γ (120572119899 + 120573 minus 119899 minus 120582)


sdot 120593 (119905) 119889119905

= (minus1)119899(

119889

119889119909)

119899minus1

int

infin

119909


times 119864120574120575119902


120572] sdot 120593 (119905) 119889119905


(119863120582

minusE120574120575119902

120572120573119901119908infin120593) (119909)

= (minus1)119899(minus1)119899int

infin

119909


119864120574120575119902

120572120573minus120582119901[119908(119905 minus 119909)

120572] 120593 (119905) 119889119905

= (E120574120575119902

120572120573minus120582119901119908infin120593) (119909)

(44)

References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Generalized Mittag-Leffler Function ...downloads.hindawi.com/journals/jmath/2013/821762.pdf · function; its various properties including di erentiation, Laplace,Beta,andMellintransforms,andgeneralizedhyper-geometric