Research ArticleGeneralized Fractional Integral Formulas forthe 119896-Bessel Function

D L Suthar 1 and Mengesha Ayene2

1Department of Mathematics Wollo University PO Box 1145 Dessie Ethiopia2Department of Physics Wollo University PO Box 1145 Dessie Ethiopia

Correspondence should be addressed to D L Suthar dlsuthargmailcom

Received 13 May 2018 Accepted 1 August 2018 Published 5 September 2018

Academic Editor Sunil D Purohit

Copyright copy 2018 D L Suthar and Mengesha Ayene This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

The aim of this paper is to deal with two integral transforms involving the Appell function as their kernels We prove somecompositions formulas for generalized fractional integrals with 119896-Bessel functionThe results are expressed in terms of generalizedWright type hypergeometric function and generalized hypergeometric series Also the authors presented some related assertionfor Saigo Riemann-Liouville type and Erdelyi-Kober type fractional integral transforms

1 Introduction and Preliminaries

The generalization of 119896-Bessel function is defined in Mondal[1] as

119882119896120585119888 (119911) =infinsum119899=0

(minus119888)119899Γ119896 (119899119896 + 120585 + 119896) 119899 (

1199112)2119899+120585119896 (1)

where 120585 gt minus1 119896 gt 0 and 119888 isin R and Γ119896(119911) is the 119896-gammafunction defined in Daaz and Pariguan [2] as

Γ119896 (119911) = intinfin0

119905119911minus1119890minus119905119896119896119889119905 119911 isin C (2)

By inspection the following relation holds

Γ119896 (119911 + 119896) = 119911Γ119896 (119911) (3)

and

Γ119896 (119911) = 119896(119911119896)minus1Γ (119911119896) (4)

If 119888 = 1 and 119896 997888rarr 1 then the generalized 119896-Bessel functiondefined in (1) reduces to the classical Bessel function 119869Vdefined in Erdelyi [3] For more details regarding 119896-Besselfunction and its properties see [4ndash9]

Here we establish various generalized fractional integralformulas for the 119896-Bessel function For this we recall here the

definition of generalized fractional integration operators ofarbitrary order involving the Appell function 1198653 ([10] p393eq (412) and (413)) as in the kernel A generalization of thehypergeometric fractional integrals for 120592 1205921015840 120583 1205831015840 120578 isin C andR(120578) isin C is established by Marichev [11] as follows

(119868(12059212059210158401205831205831015840120578)0+ 119891) (119909) = 119909minus120592Γ (120578) int119909

0(119909 minus 119905)120578minus1

sdot 119905minus12059210158401198653 (120592 1205921015840 120583 1205831015840 120578 1 minus 119905119909 1 minus 119909

119905 )119891 (119905) 119889119905(5)

(119868(12059212059210158401205831205831015840120578)minus 119891) (119909) = 119909minus1205921015840Γ (120578) intinfin

119909(119905 minus 119909)120578minus1

sdot 119905minus1205921198653 (120592 1205921015840 120583 1205831015840 120578 1 minus 119909119905 1 minus 119905

119909)119891 (119905) 119889119905(6)

In (5) and (6) 1198653 denotes the Appell function (also well-known as Horn function) which is established by Srivastavaand Karlsson [12]

1198653 (120592 1205921015840 120583 1205831015840 120578 119909 119910)

= infinsum119901119902=0

(120592)119901 (1205921015840)119902 (120583)119901 (1205831015840)119902(120578)119901+119902 119901119902 119909119901119910119902max |119909| 10038161003816100381610038161199101003816100381610038161003816

lt 1

(7)

HindawiJournal of MathematicsVolume 2018 Article ID 5198621 8 pageshttpsdoiorg10115520185198621

2 Journal of Mathematics

The properties of this function are studied in Olver et al ([13]P (412-415)) Further relation for the Gauss hypergeometricfunctions exists as follows

1198653 (120592 120577 minus 120592 120583 120577 minus 120583 120577 119909 119910)= 21198651 (120592 120583 120577 120577 119909 + 119910 minus 119909119910) (8)

The generalized hypergeometric type function 119901119865119902(119911) isrepresent in Erdelyi [14] as

119901119865119902 (119911) = 119901119865119902 [[(1205921) (1205922) sdot sdot sdot (120592119901)(1205831) (1205832) sdot sdot sdot (120583119902) 119911]

]= infinsum119899=0

(1205921)119899 (1205922)119899 sdot sdot sdot (120592119901)119899(1205831)119899 (1205832)119899 sdot sdot sdot (120583119902)119899

119911119899119899

(9)

where 120592119894 120583119895 isin C 119887119895 = 0 minus1 minus2 sdot sdot sdot 119894 = 1 2 sdot sdot sdot 119901 119895 =1 2 sdot sdot sdot 119902 and (119911)119899 is the Pochhammer symbols Now thegamma function is defined as

Γ (119911) = intinfin0

119905119911minus1119890minus119905119889119905 120582 isin C (10)

Γ (119911 + 119898) = (119911)119898 (Γ (119911)) 119911 isin C (11)

and beta function is termed as

119861 (119909 119910) = int10119905119909minus1 (1 minus 119905)119910minus1 119889119905 (12)

The following series is defined in the Wright type hypergeo-metric function (see [15ndash17]) as

119901Ψ119902 [[(120592119894 119860 119894)1119901(120583119895 119861119895)1119902 119911]

]= 119901Ψ119902 (119911)

= infinsum119899=0

Γ (1205921 + 1198601119899) sdot sdot sdot Γ (120592119901 + 119860119901119899)Γ (1205831 + 1198611119899) sdot sdot sdot Γ (120583119902 + 119861119902119899)

119911119899119899

(13)

where 120583119904 and 120592119903 are positive real numbers such that

1 + 119902sum119904=1

120583119904 minus119901sum119903=1

120592119903 gt 0 (14)

Equation (13) differs from the generalized hypergeometricfunction 119901119865119902(119911) defined in (9) only by a constant multiplier

The generalized hypergeometric function 119901119865119902(119911) is a specialcase of 119901Ψ119902(119911) for 119860 119894 = 119861119895 = 1 where 119894 = 1 2 sdot sdot sdot 119901 and119895 = 1 2 sdot sdot sdot 119902

1prod119902119895=1Γ (120583119895)119901

119865119902 [[(1205921) sdot sdot sdot (120592119901)(1205831) sdot sdot sdot (120583119902) 119911]

]= 1

prod119901119894=1Γ (120592119894) 119901Ψ119902 [[(120592119894 1)1119901(120583119895 1)1119902 119911]

]

(15)

For various properties of this function see [18]

2 Representation of Generalized FractionalIntegrals in terms of Wright Functions

Marichev-Saigo-Maeda integrals operators were generaliza-tion of Saigo fractional integral operators [19] In additiontheir properties have been studied by Saigo and Maeda [10]Considering this the left-hand side and right-hand side oftypes (5) and (6) for a power function are as follows

Lemma 1 (a) If R(120578) gt 0R(120575) gt max0R(120592 + 1205921015840 + 120583 minus120578)R(1205921015840 minus 1205831015840) then(119868(12059212059210158401205831205831015840120578)0+ 119909120575minus1) (119909)

= Γ (120575) Γ (120575 + 1205831015840 minus 1205921015840) Γ (120575 + 120578 minus 120592 minus 120592 minus 120583)Γ (120575 + 1205831015840) Γ (120575 + 120578 minus 1205921015840 minus 120583) Γ (120575 + 120578 minus 120592 minus 1205921015840)119879

(16)

(b) IfR(120578) gt 0R(120575) lt 1 +minR(minus120583)R(120592 + 1205921015840 minus 120578)R(120592 +1205831015840 minus 120578) then(119868(12059212059210158401205831205831015840 120578)minus 119909120575minus1) (119909)

= Γ (1 minus 120575 minus 120578 + 120592 + 1205921015840) Γ (1 minus 120575 minus 120583) Γ (1 minus 120575 + 120592 + 1205831015840 minus 120578)Γ (1 minus 120575) Γ (1 minus 120575 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 minus 120575 + 120592 minus 120583) 119879

(17)

where 119879 = 119909120575+120578minus120592minus1205921015840minus1The left-hand side generalized fractional integration (5) of

the 119896-Bessel functions (1) is given by the following result

Theorem2 Letting 120592 1205921015840 120583 1205831015840 120578 120582 120585 isin C be such thatR(120578) gt0R(120585) gt minus1R(120582+ V+2119899119896) ge R(120582+ 120585) gt max[0R(120592+ 1205921015840 +120583 minus 120578)R(1205921015840 minus 1205831015840)] then the following formula holds

(119868(12059212059210158401205831205831015840 120578)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909)= 119909((120582+120585)119896)+120578minus120592minus1205921015840minus1

(2119896)(120582119896)

times3Ψ4 [[[[

(120582 + 120585119896 2) (120582 + 120585

119896 minus 120592 minus 120592 minus 120583 + 120578 2) (120582 + 120585119896 + 1205831015840 minus 1205921015840 2)

(120582 + 120585119896 + 1205831015840 2) (120582 + 120585

119896 minus 120592 minus 1205921015840 + 120578 2) (120582 + 120585119896 minus 1205921015840 minus 120583 + 120578 2) ( V119896 + 1 1)

100381610038161003816100381610038161003816100381610038161003816minus11988811990924119896

]]]]

(18)

Journal of Mathematics 3

Proof Using (1) and writing the function in the series formthe left-hand side of (18) leads to

(119868(12059212059210158401205831205831015840120578)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909)= infinsum119899=0

(minus119888)119899 (12)(120585119896)+2119899Γ119896 (120585 + 119896 + 119899119896) 119899

times (119868(12059212059210158401205831205831015840120578)0+ 119905((120582+120585)119896)+2119899minus1) (119909) (19)

Now upon using the image formula (16) which isvalid under the condition declared with Theorem 2 weget

(119868(12059212059210158401205831205831015840120578)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909)= 119909((120582+120585)119896)minus120592minus1205921015840+120578minus1

2120585119896infinsum119899=0

Γ ((120582 + 120585) 119896 + 2119899)Γ ((120582 + 120585) 119896 + 1205831015840 + 2119899)

times Γ ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578 + 2119899) Γ ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840 + 2119899)Γ ((120582 + 120585) 119896 minus 120592 minus 1205921015840 + 120578 + 2119899) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120583 + 120578 + 2119899) Γ (120585119896 + 1 + 119899) 119896120585119896

(minus1198881199092)119899(4119896)119899 119899

(20)

Using the definition of (15) in the right-hand side of (20) wearrive at the result (18)

Special Cases of Theorem 2(i) If we set 1205921015840 = 0 120583 = minus120574 120578 = 120592 and replace120592 by 120592 + 120583 in (18) then we get the following corollary

relating to left-hand sided Saigo fractional integral operator([19 20])

Corollary 3 Let 120592 120583 120574 120582 120585 isin C R(120585) gt minus1 and R(120582 +120585 + 2119899119896) ge R(120582 + 120585) gt max[0R(120583 minus 120574)] then the followingformula holds

(119868(120592120583120574)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120585119896)+(120582119896)minus120583minus1(2119896)(120585119896) times 2Ψ3 [[[

[(120582 + 120585

119896 2) (120582 + 120585119896 + 120574 minus 120583 2)

(120582 + 120585119896 minus 120583 2) (120582 + 120585

119896 + 120592 + 120574 2) ( V119896 + 1 1) 100381610038161003816100381610038161003816100381610038161003816minus

11988811990924119896

]]]] (21)

(ii) If we set 120583 = minus120592 in (21) then we get the subsequentcorollary relating to left-sided Riemann-Liouville type inte-gral operator

Corollary 4 Let 120592 120582 120585 isin C be such that R(120592) gt 0R(120582 + 120585) gt 0 R(120585) gt minus1 then the following resultholds

(119868(120592)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909)

= 119909(120585119896)+(120582119896)+120592minus1(2119896)(120585119896)

times 1Ψ2 [[[[

(120582 + 120585119896 2)

(120582 + 120585119896 + 120592 2) ( V119896 + 1 1)

100381610038161003816100381610038161003816100381610038161003816minus11988811990924119896

]]]]

(22)

(iii) If we set 120583 = 0 in (21) then we get the subsequentcorollary relating to left-sided Erdelyi-Kober type integraloperator

Corollary 5 Let 120592 120574 120582 120585 isin C be such that R(120592) gt 0 R(120582 +120585 + 120574) gt 0R(120585) gt minus1 then the following formula holds

(119870+(120592120574)119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120585119896)+(120582119896)minus1(2119896)(120585119896)

times 1Ψ2 [[[[

(120582 + 120585119896 + 120574 2)

(120582 + 120585119896 + 120574 + 120592 2) ( 120585

119896 + 1 1) 100381610038161003816100381610038161003816100381610038161003816minus

11988811990924119896

]]]](23)

Theorem6 Letting 120592 1205921015840 120583 1205831015840 120578 120582 120585 isin C be such thatR(120578) gt0R(120585) gt minus1 R(120582 minus 120585 minus 2119899 minus 1) le 1 + R(120582 minus 120585 minus 1) lt1+minR(minus120583)R(120592+1205921015840minus120578)R(120592+1205831015840minus120578) then the followingformula holds

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909)

4 Journal of Mathematics

= 119909((120582minus120585)119896)minus120592minus1205921015840+120578minus1(2119896)(120585119896)

times 3Ψ4 [[[[

(1 + 120585 minus 120582119896 minus 120583 2) (1 + 120585 minus 120582

119896 + 120592 + 1205921015840 minus 120578 2) (1 + 120585 minus 120582119896 + 120592 + 1205831015840 minus 120578 2)

(120585119896 + 1 1) (1 + 120585 minus 120582

119896 2) (1 + 120585 minus 120582119896 + 120592 + 1205921015840 + 1205831015840 minus 120578 2) (1 + 120585 minus 120582

119896 + 120592 minus 120583 2) 1003816100381610038161003816100381610038161003816minus

11988841198961199092

]]]]

(24)

Proof Using (2) and writing the function in the series formthe left-hand side of (24) leads to

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909)

= infinsum119899=0

(minus119888)119899 (12)(120585119896)+2119899Γ119896 (120585 + 119896 + 119899119896) 119899

times (119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus(120585119896)minus2119899minus1) (119909) (25)

Now upon using the image formula (17) which is validunder the conditions declared with Theorem 6 weget

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909)

= 119909(120582119896)minus(120585119896)minus120592minus1205921015840+120578minus1(2119896)(]119896)

times infinsum119899=0

Γ (1 + (120585 minus 120582) 119896 + 120592 + 1205921015840 minus 120578 + 2119899) Γ (1 + (120585 minus 120582) 119896 + 120592 + 1205831015840 minus 120578 + 2119899)Γ (1 + (120585 minus 120582) 119896 + 2119899) Γ (1 + (120585 minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578 + 2119899) Γ (1 + (120585 minus 120582) 119896 + 120592 minus 120583 + 2119899)

times Γ (1 + (120585 minus 120582) 119896 minus 120583 + 2119899)Γ (120585119896 + 1 + 119899)

(minus119888)119899(41198961199092)119899 119899

(26)

Using the definition of (15) in the right-hand side of (26) wearrive at the result (24)

Special Cases of Theorem 6(iv) If we substitute 1205921015840 = 0 120583 = minus120574 120578 = 120592 and replace120592 by 120592 + 120583 in (24) then we get the subsequent corollary

relating to right-hand sided Saigo fractional integral operator[19]

Corollary 7 Letting 120592 120583 120574 120582 120585 isin C andR(120585) gt minus1R(120592) gt0R(120582minus120585) lt 1+min[R(120583)R(120574)] then the following formulaholds

(119868(120592120583120574)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909((120582minus120585)119896)minus120583minus1

(2119896)(120585119896)

times 2Ψ3 [[[[

(1 + 120583 + 120585 minus 120582119896 2) (1 + 120585 minus 120582

119896 + 120574 2) (1 + 120585 minus 120582

119896 2) (1 + 120583 + 120592 + 120574 + 120585 minus 120582119896 2) (120585

119896 + 1 1) 1003816100381610038161003816100381610038161003816minus

11988841199092

]]]]

(27)

(v) If we set 120583 = minus120592 in (27) then we get the followingcorollary relating to right-sided Weyl fractional type integraloperator

Corollary 8 Let 120592 120582 120585 isin C be such that R(120592) gt 0R(120585) gt minus1 R(120592) lt 1 minus R(120582 minus 120585) then the following resultholds

(119868(120592)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120585119896)minus(120582119896)+120592minus1

(2119896)(120585119896)

times 1Ψ2 [[[[

(1 minus 120592 + 120585 minus 120582119896 2)

(120585 minus 120582119896 + 1 2) (120585

119896 + 1 1) 1003816100381610038161003816100381610038161003816minus

11988841198961199092

]]]](28)

Journal of Mathematics 5

(vi) If we set 120583 = 0 in (27) then we get the subsequentcorollary relating to right-hand side of Erdelyi-Kober frac-tional type integral operator

Corollary 9 Let 120592 120574120582 120585 isin C be such thatR(120585) gt minus1R(120592) gt0 R(120582 + 120585) lt 1 + min[0R(120574)] then the following formulaholds

(119870minus(120592120574)119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120585119896)minus(120582119896)minus1

(2119896)(120585119896)

times 1Ψ2 [[[[

(1 + 120585 minus 120582119896 + 120574 2)

(1 + 120585 minus 120582119896 + 120592 + 120574 2) ( 120585

119896 + 1 1) 1003816100381610038161003816100381610038161003816minus

11988841198961199092

]]]](29)

3 Representation in Terms of GeneralizedHypergeometric Function

In this part we introduce the generalized fractionalintegrals of 119896-Bessel function in terms of generalized

hypergeometric function We consider the followingwell-known results

Γ (2119911) = 22119911minus1radic120587 Γ (119911) Γ (119911 + 1

2) 119911 isin C (30)

and

(119911)2119899 = 22119899 (1199112)119899 (

1199112 + 1

2)119899 119911 isin C 119899 isin N (31)

We represent the following theorems containing the general-ized hypergeometric function

Theorem 10 Let 120592 1205921015840 120583 1205831015840 120578 120582 120585 isin C be such thatR(120578) gt 0R(120585) gt minus1 R(120582 + 120585 + 2119899119896) ge R(120582 + 120585) gt max[0R(120592 +1205921015840 + 120583 minus 120578)R(1205921015840 minus 1205831015840)] and let (120582 + 120585)119896 (120582 + 120585)119896 minus120592 minus 120592 minus 120583 + 120578 = 0 minus1 sdot sdot sdot then the following formulaholds

(119868(12059212059210158401205831205831015840120578)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909((120582+120585)119896)minus120592minus1205921015840+120578minus1(2119896)(120585119896) times Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578) Γ ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)

Γ ((120582 + 120585) 119896 + 1205831015840) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120592 + 120578) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120583 + 120578) Γ (120585119896 + 1)

times 61198657 [[[[

120582 + 1205852119896 120582 + 120585

2119896 + 12

120582 + 1205852119896 + 120578 minus 120592 minus 120592 minus 120583

2 120582 + 1205852119896 + 120578 minus 120592 minus 120592 minus 120583 + 1

2 120582 + 1205852119896 + 1205831015840 minus 1205921015840

2 120582 + 1205852119896 + 1205831015840 minus 1205921015840 + 1

2 120585119896 + 1 120582 + 120585

2119896 + 12058310158402 120582 + 120585

2119896 + 1205831015840 + 12 120582 + 120585

2119896 + 120578 minus 120592 minus 12059210158402 120582 + 120585

2119896 + 120578 minus 120592 minus 1205921015840 + 12 120582 + 120585

2119896 + 120578 minus 1205921015840 minus 1205832 120582 + 120585

2119896 + 120578 minus 1205921015840 minus 120583 + 12

100381610038161003816100381610038161003816100381610038161003816minus11988811990924119896

]]]](32)

Proof Note that 61198657 defined in (32) exit as the series isabsolutely convergent Now using (11) with 119911 = 120585119896 + 1 and

(20) and applying (31) with 119911 being replaced by (120582 + 120585)119896(120582 + 120585)119896 minus 120592 minus 120592 minus 120583 + 120578 and (120582 + 120585)119896 + 1205831015840 minus 1205921015840 we have

(119868(12059212059210158401205831205831015840120578)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909((120582+120585)119896)minus120592minus1205921015840+120578minus1(2119896)(120585119896)

times infinsum119899=0

Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578) Γ ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)Γ ((120582 + 120585) 119896 + 1205831015840) Γ ((120582 + 120585) 119896 minus 120592 minus 1205921015840 + 120578) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120583 + 120578) Γ (120585119896 + 1)

times ((120582 + 120585) 119896)2119899 ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578)2119899 ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)2119899((120582 + 120585) 119896 + 1205831015840)2119899 ((120582 + 120585) 119896 minus 120592 minus 1205921015840 + 120578)2119899 ((120582 + 120585) 119896 + 120578 minus 1205921015840 minus 120583)2119899 (120585119896 + 1)119899

(minus1198881199092)119899(4119896)119899 119899

= 119909((120582+120585)119896)minus120592minus1205921015840+120578minus1(2119896)(120585119896)

Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578) Γ ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)Γ ((120582 + 120585) 119896 + 1205831015840) Γ ((120582 + 120585) 119896 minus 120592 minus 1205921015840 + 120578) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120583 + 120578) Γ (120585119896 + 1)

times infinsum119899=0

((120582 + 120585) 2119896)119899 ((120582 + 120585) 2119896 + 12)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 120592 minus 120583) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 120592 minus 120583 + 1) 2)119899(120585119896 + 1)119899 ((120582 + 120585) 2119896 + 12058310158402)119899 ((120582 + 120585) 2119896 + (1205831015840 + 1) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 1205921015840) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 1205921015840 + 1) 2)119899

times ((120582 + 120585) 2119896 + (1205831015840 minus 1205921015840) 2)119899((120582 + 120585) 2119896 + (1205831015840 minus 1205921015840 + 1) 2)

119899((120582 + 120585) 2119896 + (120578 minus 1205921015840 minus 120583) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 1205921015840 minus 120583 + 1) 2)119899(minus1198881199092)119899(4119896)119899 119899

(33)

Thus in accordance with (9) we get the required result(32)

Corollary 11 Let 120592 120583 120574 120582 120585 isin C be such that R(120592) gt 0R(120585) gt minus1R(120582 + 120585) gt max[0R(120583 minus 120574)] and let (120582 + 120585)119896

6 Journal of Mathematics

(120582 + 120585)119896 + 120574 minus 120583 = 0 minus1 sdot sdot sdot then the following resultholds

(119868(120592120583120574)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120585119896)+(120582119896)minus120583minus1(2119896)(120585119896)

Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 + 120574 minus 120583)Γ (V119896 + 1) Γ ((120582 + 120585) 119896 minus 120583) Γ ((120582 + 120585) 119896 + 120592 + 120574)

times41198655 [[[

120582 + 1205852119896 120582 + 120585

2119896 + 12

120582 + 1205852119896 + 120574 minus 120583

2 120582 + 1205852119896 + 1 + 120574 minus 120583

2 V119896 + 1 120582 + 120585

2119896 minus 1205832

120582 + 1205852119896 minus 120583 + 1

2 120582 + 120585119896 + 120592 + 120574

2 120582 + 1205852119896 + 1 + 120592 + 120582

2 100381610038161003816100381610038161003816100381610038161003816minus

11988811990924119896 ]]

]

(34)

Corollary 12 Let 120592 120582 120585 isin C be such that R(120592) gt 0 R(120582 +120585) gt 0 R(120585) gt minus1 and 120582119896 + 120585119896 = 0 minus1 sdot sdot sdot then thefollowing result holds

(119868(120592)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120582119896)+(120585119896)+120592minus1(2119896)(120585119896)

sdot Γ ((120582 + 120585) 119896)Γ (1 + 120585119896) Γ ((120582 + 120585) 119896 minus 120583)

times 21198653 [[[

120582 + 1205852119896 120582 + 120585

2119896 + 12 120585

119896 + 1 120582 + 1205852119896 minus 120583

2 120582 + 1205852119896 minus 120583 + 1

2 100381610038161003816100381610038161003816100381610038161003816minus

11988811990924119896 ]]

](35)

Corollary 13 Let 120592 120574 120585 120582 isin C be such thatR(120592) gt 0R(120582 +120585) gt 0R(120585) gt minus1 and let 120582119896 + 120585119896 + 120574 minus 120583 = 0 minus1 sdot sdot sdot thenthe following result holds

(119870+(120592120574)119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120585119896)+(120582119896)minus1(2119896)(120585119896)

sdot Γ ((120582 + 120585) 119896 + 120574)Γ (1 + 120585119896) Γ ((120582 + 120585) 119896 + 120592 + 120574)

times21198653 [[[

120582 + 1205852119896 + 120574

2 120582 + 1205852119896 + 120574 + 1

2120585119896 + 1 120582 + 120585

119896 + 120574 + 1205922 120582 + 120585

2119896 + 1 + 120592 + 1205742

100381610038161003816100381610038161003816100381610038161003816minus11988811990924119896 ]]

]

(36)

Theorem 14 Letting 120592 1205921015840 120583 1205831015840 120578 120582 V isin C be such that 119896 gt 0R(120574) gt 0R(]) gt minus1R(120582 minus V minus 2119899 minus 1) le 1 +R(120582 minus V minus 1) lt1+minR(minus120583)R(120592+ 1205921015840 minus120578)R(120592+ 1205831015840 minus120578) then there holdsthe following formula

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896V119888 (1119905 )) (119909) = 119909((120582minus])119896)+120578minus120592minus1205921015840minus1

(2119896)(]119896) times Γ (1 + (] minus 120582) 119896 minus 120583) Γ (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840) Γ (1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578)Γ (V119896 + 1) Γ (1 + (] minus 120582) 119896) Γ (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 + 120592 minus 120583)

times61198657 [[[[

1 + 120592 + 1205921015840 minus 1205782 + ] minus 120582

2119896 1 + ] minus 1205822119896 + 120592 + 1205921015840 minus 120578

2 1 + 120592 + 1205831015840 minus 1205782 + ] minus 120582

2119896 1 + ] minus 1205822119896 + 120592 + 1205831015840 minus 120578

2 1 minus 1205832 + ] minus 120582

2119896 1 + ] minus 1205822119896 minus 120583

2 V119896 + 1 1 + 120592 + 1205921015840 + 1205831015840 minus 120578

2 + ] minus 1205822119896 1 + ] minus 120582

2119896 + 120592 + 1205921015840 + 1205831015840 minus 1205782 12 + ] minus 120582

2119896 1 + ] minus 1205822119896 1 + 120592 minus 120583

2 + ] minus 1205822119896 1 + ] minus 120582

2119896 + 120592 minus 1205832

1003816100381610038161003816100381610038161003816minus119888

41198961199092]]]](37)

Proof Using (11) with 119911 = V119896 + 1 and (26) and applying (31)with 119911 being replaced by (]minus120582)119896minus120583+1 (]minus120582)119896+120592+1205831015840minus120578+1and (] minus 120582)119896 minus 120578 + 120592 + 1205921015840 + 1 we have

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896V119888 (1119905 )) (119909) = 119909((120582minus])119896)+120578minus120592minus1205921015840minus1

(2119896)(]119896)

times infinsum119899=0

Γ (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840) Γ (1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 minus 120583)Γ (1 + (] minus 120582) 119896) Γ (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 + 120592 minus 120583) Γ (V119896 + 1)

times (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840)2119899

(1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578)2119899

(1 + (] minus 120582) 119896 minus 120583)2119899(1 + (] minus 120582) 119896)2119899 (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578)2119899 (1 + (] minus 120582) 119896 + 120592 minus 120583)2119899 (V119896 + 1)119899

(minus119888)119899(41198961199092)119899 119899

= 119909((120582minus])119896)+120578minus120592minus1205921015840minus1(2119896)(]119896)

Γ (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840) Γ (1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 minus 120583)Γ (1 + (] minus 120582) 119896) Γ (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 + 120592 minus 120583) Γ (V119896 + 1)

times infinsum119899=0

(12 + (] minus 120582) 2119896 + (120592 + 1205921015840 minus 120578) 2)119899(1 + (] minus 120582) 2119896 + (120592 + 1205921015840 minus 120578) 2)

119899(V119896 + 1)119899 (12 + (] minus 120582) 2119896 + (120592 + 1205921015840 + 1205831015840 minus 120578) 2)119899 (1 + (] minus 120582) 2119896 + (120592 + 1205921015840 + 1205831015840 minus 120578) 2)119899

Journal of Mathematics 7

times (12 + (] minus 120582) 2119896 + (120592 + 1205831015840 minus 120578) 2)119899(1 + (] minus 120582) 2119896 + (120592 + 1205831015840 minus 120578) 2)

119899(12 + (] minus 120582) 2119896 minus 1205832)119899 (1 + (] minus 120582) 2119896 minus 1205832)119899

(12 + (] minus 120582) 2119896)119899 (1 + (] minus 120582) 2119896)119899 (1 + (] minus 120582) 2119896 + (120592 minus 120583) 2)119899 (12 + (] minus 120582) 2119896 + (120592 minus 120583) 2)119899sdot (minus119888)119899(41198961199092)119899 119899

(38)

Thus in accordance with (9) we get the required result (37)

Corollary 15 Let 120592 120583 120574 120585 120582 isin C and 119896 gt 0 be such thatR(120592) gt 0R(120585) gt minus1 R(V) gt minus1R(120592) gt 0R(120582 minus

V) lt 1 + min[R(120583)R(120574)] and let (120583 minus 120582)119896 + 120585119896 +1 120574 minus 120582119896 + 120585119896 + 1 = 0 minus1 sdot sdot sdot then the following resultholds

(119868(120592120583120574)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909)

= 119909(120582119896)minus(120585119896)minus120583minus1(2119896)(120585119896) times Γ ((120585 minus 120582) 119896 + 120574 + 1) Γ ((120585 minus 120582) 119896 + 120583 + 1)

Γ (120585119896 + 1) Γ (1 + (120585 minus 120582) 119896) Γ ((120585 minus 120582) 119896 + 120592 + 120583 + 120578 + 1)

times41198655 [[[

120583 + 12 + 120585 minus 120582

2119896 120583 + 22 + 120585 minus 120582

2119896 120574 + 12 + 120585 minus 120582

2119896 120574 + 22 + 120585 minus 120582

2119896 V119896 + 1 12 + 120585 minus 120582

2119896 1 + 120585 minus 1205822119896 120592 + 120583 + 120574 + 1

2 + 120585 minus 120582119896 120592 + 120574 + 2

2 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]]

(39)

Corollary 16 Let 120592 120574 120585 120582 isin C and 119896 gt 0 be such that 0 ltR(120592) lt 1minusR(120582minus120585)R(120585) gt minus1 and let120582119896minus120585119896+120592 = 1 2 sdot sdot sdot then the following result holds

(119868(120592)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120582119896)minus(120585119896)+120592minus1

(2119896)(120585119896)sdot Γ (1 minus 120592 + (120585 minus 120582) 119896)Γ (120585119896 + 1) Γ (1 + (120585 minus 120582) 119896)

times 21198653 [[[

1 minus 1205832 + 120585 minus 120582

2119896 2 minus 1205922 + 120585 minus 120582

2119896 120585119896 + 1 12 + 120585 minus 120582

2119896 1 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]](40)

Corollary 17 Let 120592 120574 120585 120582 isin C and 119896 gt 0 be such thatR(120585) gtminus1R(120592) gt 0R(120582+120585) lt 1+max[0R(120574)] and let 120582119896minus120585119896minus120574 = 1 2 sdot sdot sdot then the following formula holds

(119870minus(120592120574)119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120582119896)minus(120585119896)minus1

(2119896)(120585119896)Γ ((120585 minus 120582) 119896 + 120574 + 1)

Γ (1 + (120585 minus 120582) 119896) Γ (120585119896 + 1)

times 21198653 [[[

120574 + 12 + 120585 minus 120582

2119896 120574 + 22 + 120585 minus 120582

2119896 120585119896 + 1 120592 + 120574 + 1

2 + 120585 minus 120582119896 120592 + 120574 + 2

2 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]]

(41)

4 Concluding Remark

MSM fractional integral operators have advantage that theygeneralize theR-LWeyl Erdelyi-Kober and Saigorsquos fractionalintegral operators therefore several authors called this ageneral operator So we conclude this paper by emphasizingthat many other interesting image formulas can be derivedas the specific cases of our leading results Theorems 2 and6 involving familiar fractional integral operators as abovesaid Further the generalized Bessel function defined in (1)

possesses the lead that a number of Bessel functions trigono-metric functions and hyperbolic functions happen to bethe particular cases of this function Some special cases ofintegrals involving generalized Bessel function have beenexplored in the literature by a number of authors ([20ndash26])with different arguments Therefore results presented in thispaper are easily converted in terms of a comparable typeof novel interesting integrals with diverse arguments aftervarious suitable parametric replacements

8 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] S R Mondal Representation Formulae and Monotonicity ofthe Generalized k-Bessel Functions 2016

[2] Diaz and E Pariguan ldquoOn hypergeometric functions andPochhammer k-symbolrdquo Divulgaciones Matematicas vol 15no 2 pp 179ndash192 2007

[3] H SatohHigher Transcendental Functions vol 2McGraw-HillNewYork Toronto London 1953

[4] P AgarwalM Chand J Choi andG Singh ldquoCertain fractionalintegrals and image formulas of generalized k-Bessel functionrdquoCommunications of the KoreanMathematical Society vol 33 no2 pp 423ndash436 2018

[5] M Chand P Agarwal and Z Hammouch ldquoCertain SequencesInvolving Product of k-Bessel Functionrdquo International Journalof Applied and Computational Mathematics vol 4 no 4 4101pages 2018

[6] K S Gehlot ldquoDifferential Equation of K -Bessels Functions andits Propertiesrdquo Nonl Analysis and Differential Equations vol 2no 2 pp 61ndash67 2014

[7] K S Gehlot ldquoRecurrence Relations of K -Bessels functionrdquoThaiJ Math

[8] K S Gehlot and S D Purohit ldquoIntegral representations of thek-Besselrsquos functionrdquo Honam Mathematical Journal vol 38 no1 pp 17ndash23 2016

[9] G Singh P AgarwalM Chand and S Jain ldquo Certain fractionalkinetic equations involving generalized rdquo Transactions of ARazmadze Mathematical Institute 2018

[10] M Saigo and N Maeda ldquoMore generalization of fractionalcalculusrdquo in TransformMethods and Special Functions pp 386ndash400 Bulgarian Acad Sci Sofia 1998

[11] O I Marichev ldquoVolterra equation of Mellin convolution typewith a Horn function in the kernelrdquo Izvestiya Akademii NaukBSSR Seriya Fiziko-Matematicheskikh Nauk vol 1 pp 128-1291974 (Russian)

[12] H M Srivastava and P W Karlsson Multiple Gaussian hyper-geometric series Ellis Horwood Series Mathematics and itsApplications Ellis Horwood Ltd Chichester Halsted Press[John Wiley amp Sons Inc] New York NY USA 1985

[13] F W J Olver DW Lozier R F Boisvert and CW Clark EdsNIST Handbook of Mathematical Functions National Instituteof Standards and Technology USA Cambridge University PressGaithersburg Md 2010

[14] A Erdersquolyi W Magnus F Oberhettinger and F G Tri-comi Higher Transcendental Functions vol 1 McGraw-HillNewYork Toronto London 1953

[15] E M Wright ldquoThe asymptotic expansion of the generalizedhypergeometric functionrdquo Journal of the London MathematicalSociety vol 1-10 no 4 pp 286ndash293 1935

[16] E M Wright ldquoThe asymptotic expansion of the generalizedhypergeometric functionrdquo Proceedings of the London Mathe-matical Society vol 46 no 2 pp 389ndash408 1940

[17] E M Wright ldquoThe asymptotic expansion of integral functionsdefined by Taylor seriesrdquo Philosophical Transactions of the RoyalSociety A Mathematical Physical amp Engineering Sciences vol238 pp 423ndash451 1940

[18] A A Kilbas M Saigo and J J Trujillo ldquoOn the generalizedWright functionrdquo Fractional Calculus and Applied Analysis vol5 no 4 pp 437ndash460 2002

[19] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash1431978

[20] A A Kilbas and N Sebastian ldquoGeneralized fractional integra-tion of Bessel function of the first kindrdquo Integral Transforms andSpecial Functions vol 19 no 11-12 pp 869ndash883 2008

[21] H Amsalu and D L Suthar ldquoGeneralized Fractional IntegralOperators Involving Mittag-Leffler Functionrdquo Abstract andApplied Analysis vol 2018 Article ID 7034124 8 pages 2018

[22] S R Mondal and K S Nisar ldquoMarichev-Saigo-Maeda Frac-tional Integration Operators Involving Generalized BesselFunctionsrdquo Mathematical Problems in Engineering vol 2014Article ID 274093 11 pages 2014

[23] K S Nisar D L Suthar S D Purohit and M AldhaifallahldquoSome unified integrals associated with the generalized Struvefunctionrdquo Proceedings of the Jangjeon Mathematical SocietyMemoirs of the Jangjeon Mathematical Society vol 20 no 2 pp261ndash267 2017

[24] S D Purohit D L Suthar and S L Kalla ldquoMarichev-Saigo-Maeda fractional integration operators of the Bessel functionsrdquoLe Matematiche vol 67 no 1 pp 21ndash32 2012

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

[26] D L Suthar and H Habenom ldquoIntegrals involving generalizedBessel-Maitland functionrdquo Journal of Science andArts no 4(37)pp 357ndash362 2016

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Journal of Mathematics 3

Proof Using (1) and writing the function in the series formthe left-hand side of (18) leads to

(119868(12059212059210158401205831205831015840120578)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909)= infinsum119899=0

(minus119888)119899 (12)(120585119896)+2119899Γ119896 (120585 + 119896 + 119899119896) 119899

times (119868(12059212059210158401205831205831015840120578)0+ 119905((120582+120585)119896)+2119899minus1) (119909) (19)

Now upon using the image formula (16) which isvalid under the condition declared with Theorem 2 weget

(119868(12059212059210158401205831205831015840120578)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909)= 119909((120582+120585)119896)minus120592minus1205921015840+120578minus1

2120585119896infinsum119899=0

Γ ((120582 + 120585) 119896 + 2119899)Γ ((120582 + 120585) 119896 + 1205831015840 + 2119899)

times Γ ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578 + 2119899) Γ ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840 + 2119899)Γ ((120582 + 120585) 119896 minus 120592 minus 1205921015840 + 120578 + 2119899) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120583 + 120578 + 2119899) Γ (120585119896 + 1 + 119899) 119896120585119896

(minus1198881199092)119899(4119896)119899 119899

(20)

Using the definition of (15) in the right-hand side of (20) wearrive at the result (18)

Special Cases of Theorem 2(i) If we set 1205921015840 = 0 120583 = minus120574 120578 = 120592 and replace120592 by 120592 + 120583 in (18) then we get the following corollary

relating to left-hand sided Saigo fractional integral operator([19 20])

Corollary 3 Let 120592 120583 120574 120582 120585 isin C R(120585) gt minus1 and R(120582 +120585 + 2119899119896) ge R(120582 + 120585) gt max[0R(120583 minus 120574)] then the followingformula holds

(119868(120592120583120574)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120585119896)+(120582119896)minus120583minus1(2119896)(120585119896) times 2Ψ3 [[[

[(120582 + 120585

119896 2) (120582 + 120585119896 + 120574 minus 120583 2)

(120582 + 120585119896 minus 120583 2) (120582 + 120585

119896 + 120592 + 120574 2) ( V119896 + 1 1) 100381610038161003816100381610038161003816100381610038161003816minus

11988811990924119896

]]]] (21)

(ii) If we set 120583 = minus120592 in (21) then we get the subsequentcorollary relating to left-sided Riemann-Liouville type inte-gral operator

Corollary 4 Let 120592 120582 120585 isin C be such that R(120592) gt 0R(120582 + 120585) gt 0 R(120585) gt minus1 then the following resultholds

(119868(120592)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909)

= 119909(120585119896)+(120582119896)+120592minus1(2119896)(120585119896)

times 1Ψ2 [[[[

(120582 + 120585119896 2)

(120582 + 120585119896 + 120592 2) ( V119896 + 1 1)

100381610038161003816100381610038161003816100381610038161003816minus11988811990924119896

]]]]

(22)

(iii) If we set 120583 = 0 in (21) then we get the subsequentcorollary relating to left-sided Erdelyi-Kober type integraloperator

Corollary 5 Let 120592 120574 120582 120585 isin C be such that R(120592) gt 0 R(120582 +120585 + 120574) gt 0R(120585) gt minus1 then the following formula holds

(119870+(120592120574)119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120585119896)+(120582119896)minus1(2119896)(120585119896)

times 1Ψ2 [[[[

(120582 + 120585119896 + 120574 2)

(120582 + 120585119896 + 120574 + 120592 2) ( 120585

119896 + 1 1) 100381610038161003816100381610038161003816100381610038161003816minus

11988811990924119896

]]]](23)

Theorem6 Letting 120592 1205921015840 120583 1205831015840 120578 120582 120585 isin C be such thatR(120578) gt0R(120585) gt minus1 R(120582 minus 120585 minus 2119899 minus 1) le 1 + R(120582 minus 120585 minus 1) lt1+minR(minus120583)R(120592+1205921015840minus120578)R(120592+1205831015840minus120578) then the followingformula holds

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909)

4 Journal of Mathematics

= 119909((120582minus120585)119896)minus120592minus1205921015840+120578minus1(2119896)(120585119896)

times 3Ψ4 [[[[

(1 + 120585 minus 120582119896 minus 120583 2) (1 + 120585 minus 120582

119896 + 120592 + 1205921015840 minus 120578 2) (1 + 120585 minus 120582119896 + 120592 + 1205831015840 minus 120578 2)

(120585119896 + 1 1) (1 + 120585 minus 120582

119896 2) (1 + 120585 minus 120582119896 + 120592 + 1205921015840 + 1205831015840 minus 120578 2) (1 + 120585 minus 120582

119896 + 120592 minus 120583 2) 1003816100381610038161003816100381610038161003816minus

11988841198961199092

]]]]

(24)

Proof Using (2) and writing the function in the series formthe left-hand side of (24) leads to

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909)

= infinsum119899=0

(minus119888)119899 (12)(120585119896)+2119899Γ119896 (120585 + 119896 + 119899119896) 119899

times (119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus(120585119896)minus2119899minus1) (119909) (25)

Now upon using the image formula (17) which is validunder the conditions declared with Theorem 6 weget

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909)

= 119909(120582119896)minus(120585119896)minus120592minus1205921015840+120578minus1(2119896)(]119896)

times infinsum119899=0

Γ (1 + (120585 minus 120582) 119896 + 120592 + 1205921015840 minus 120578 + 2119899) Γ (1 + (120585 minus 120582) 119896 + 120592 + 1205831015840 minus 120578 + 2119899)Γ (1 + (120585 minus 120582) 119896 + 2119899) Γ (1 + (120585 minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578 + 2119899) Γ (1 + (120585 minus 120582) 119896 + 120592 minus 120583 + 2119899)

times Γ (1 + (120585 minus 120582) 119896 minus 120583 + 2119899)Γ (120585119896 + 1 + 119899)

(minus119888)119899(41198961199092)119899 119899

(26)

Using the definition of (15) in the right-hand side of (26) wearrive at the result (24)

Special Cases of Theorem 6(iv) If we substitute 1205921015840 = 0 120583 = minus120574 120578 = 120592 and replace120592 by 120592 + 120583 in (24) then we get the subsequent corollary

relating to right-hand sided Saigo fractional integral operator[19]

Corollary 7 Letting 120592 120583 120574 120582 120585 isin C andR(120585) gt minus1R(120592) gt0R(120582minus120585) lt 1+min[R(120583)R(120574)] then the following formulaholds

(119868(120592120583120574)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909((120582minus120585)119896)minus120583minus1

(2119896)(120585119896)

times 2Ψ3 [[[[

(1 + 120583 + 120585 minus 120582119896 2) (1 + 120585 minus 120582

119896 + 120574 2) (1 + 120585 minus 120582

119896 2) (1 + 120583 + 120592 + 120574 + 120585 minus 120582119896 2) (120585

119896 + 1 1) 1003816100381610038161003816100381610038161003816minus

11988841199092

]]]]

(27)

(v) If we set 120583 = minus120592 in (27) then we get the followingcorollary relating to right-sided Weyl fractional type integraloperator

Corollary 8 Let 120592 120582 120585 isin C be such that R(120592) gt 0R(120585) gt minus1 R(120592) lt 1 minus R(120582 minus 120585) then the following resultholds

(119868(120592)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120585119896)minus(120582119896)+120592minus1

(2119896)(120585119896)

times 1Ψ2 [[[[

(1 minus 120592 + 120585 minus 120582119896 2)

(120585 minus 120582119896 + 1 2) (120585

119896 + 1 1) 1003816100381610038161003816100381610038161003816minus

11988841198961199092

]]]](28)

Journal of Mathematics 5

(vi) If we set 120583 = 0 in (27) then we get the subsequentcorollary relating to right-hand side of Erdelyi-Kober frac-tional type integral operator

Corollary 9 Let 120592 120574120582 120585 isin C be such thatR(120585) gt minus1R(120592) gt0 R(120582 + 120585) lt 1 + min[0R(120574)] then the following formulaholds

(119870minus(120592120574)119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120585119896)minus(120582119896)minus1

(2119896)(120585119896)

times 1Ψ2 [[[[

(1 + 120585 minus 120582119896 + 120574 2)

(1 + 120585 minus 120582119896 + 120592 + 120574 2) ( 120585

119896 + 1 1) 1003816100381610038161003816100381610038161003816minus

11988841198961199092

]]]](29)

3 Representation in Terms of GeneralizedHypergeometric Function

In this part we introduce the generalized fractionalintegrals of 119896-Bessel function in terms of generalized

hypergeometric function We consider the followingwell-known results

Γ (2119911) = 22119911minus1radic120587 Γ (119911) Γ (119911 + 1

2) 119911 isin C (30)

and

(119911)2119899 = 22119899 (1199112)119899 (

1199112 + 1

2)119899 119911 isin C 119899 isin N (31)

We represent the following theorems containing the general-ized hypergeometric function

Theorem 10 Let 120592 1205921015840 120583 1205831015840 120578 120582 120585 isin C be such thatR(120578) gt 0R(120585) gt minus1 R(120582 + 120585 + 2119899119896) ge R(120582 + 120585) gt max[0R(120592 +1205921015840 + 120583 minus 120578)R(1205921015840 minus 1205831015840)] and let (120582 + 120585)119896 (120582 + 120585)119896 minus120592 minus 120592 minus 120583 + 120578 = 0 minus1 sdot sdot sdot then the following formulaholds

(119868(12059212059210158401205831205831015840120578)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909((120582+120585)119896)minus120592minus1205921015840+120578minus1(2119896)(120585119896) times Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578) Γ ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)

Γ ((120582 + 120585) 119896 + 1205831015840) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120592 + 120578) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120583 + 120578) Γ (120585119896 + 1)

times 61198657 [[[[

120582 + 1205852119896 120582 + 120585

2119896 + 12

120582 + 1205852119896 + 120578 minus 120592 minus 120592 minus 120583

2 120582 + 1205852119896 + 120578 minus 120592 minus 120592 minus 120583 + 1

2 120582 + 1205852119896 + 1205831015840 minus 1205921015840

2 120582 + 1205852119896 + 1205831015840 minus 1205921015840 + 1

2 120585119896 + 1 120582 + 120585

2119896 + 12058310158402 120582 + 120585

2119896 + 1205831015840 + 12 120582 + 120585

2119896 + 120578 minus 120592 minus 12059210158402 120582 + 120585

2119896 + 120578 minus 120592 minus 1205921015840 + 12 120582 + 120585

2119896 + 120578 minus 1205921015840 minus 1205832 120582 + 120585

2119896 + 120578 minus 1205921015840 minus 120583 + 12

100381610038161003816100381610038161003816100381610038161003816minus11988811990924119896

]]]](32)

Proof Note that 61198657 defined in (32) exit as the series isabsolutely convergent Now using (11) with 119911 = 120585119896 + 1 and

(20) and applying (31) with 119911 being replaced by (120582 + 120585)119896(120582 + 120585)119896 minus 120592 minus 120592 minus 120583 + 120578 and (120582 + 120585)119896 + 1205831015840 minus 1205921015840 we have

(119868(12059212059210158401205831205831015840120578)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909((120582+120585)119896)minus120592minus1205921015840+120578minus1(2119896)(120585119896)

times infinsum119899=0

Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578) Γ ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)Γ ((120582 + 120585) 119896 + 1205831015840) Γ ((120582 + 120585) 119896 minus 120592 minus 1205921015840 + 120578) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120583 + 120578) Γ (120585119896 + 1)

times ((120582 + 120585) 119896)2119899 ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578)2119899 ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)2119899((120582 + 120585) 119896 + 1205831015840)2119899 ((120582 + 120585) 119896 minus 120592 minus 1205921015840 + 120578)2119899 ((120582 + 120585) 119896 + 120578 minus 1205921015840 minus 120583)2119899 (120585119896 + 1)119899

(minus1198881199092)119899(4119896)119899 119899

= 119909((120582+120585)119896)minus120592minus1205921015840+120578minus1(2119896)(120585119896)

Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578) Γ ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)Γ ((120582 + 120585) 119896 + 1205831015840) Γ ((120582 + 120585) 119896 minus 120592 minus 1205921015840 + 120578) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120583 + 120578) Γ (120585119896 + 1)

times infinsum119899=0

((120582 + 120585) 2119896)119899 ((120582 + 120585) 2119896 + 12)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 120592 minus 120583) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 120592 minus 120583 + 1) 2)119899(120585119896 + 1)119899 ((120582 + 120585) 2119896 + 12058310158402)119899 ((120582 + 120585) 2119896 + (1205831015840 + 1) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 1205921015840) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 1205921015840 + 1) 2)119899

times ((120582 + 120585) 2119896 + (1205831015840 minus 1205921015840) 2)119899((120582 + 120585) 2119896 + (1205831015840 minus 1205921015840 + 1) 2)

119899((120582 + 120585) 2119896 + (120578 minus 1205921015840 minus 120583) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 1205921015840 minus 120583 + 1) 2)119899(minus1198881199092)119899(4119896)119899 119899

(33)

Thus in accordance with (9) we get the required result(32)

Corollary 11 Let 120592 120583 120574 120582 120585 isin C be such that R(120592) gt 0R(120585) gt minus1R(120582 + 120585) gt max[0R(120583 minus 120574)] and let (120582 + 120585)119896

6 Journal of Mathematics

(120582 + 120585)119896 + 120574 minus 120583 = 0 minus1 sdot sdot sdot then the following resultholds

(119868(120592120583120574)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120585119896)+(120582119896)minus120583minus1(2119896)(120585119896)

Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 + 120574 minus 120583)Γ (V119896 + 1) Γ ((120582 + 120585) 119896 minus 120583) Γ ((120582 + 120585) 119896 + 120592 + 120574)

times41198655 [[[

120582 + 1205852119896 120582 + 120585

2119896 + 12

120582 + 1205852119896 + 120574 minus 120583

2 120582 + 1205852119896 + 1 + 120574 minus 120583

2 V119896 + 1 120582 + 120585

2119896 minus 1205832

120582 + 1205852119896 minus 120583 + 1

2 120582 + 120585119896 + 120592 + 120574

2 120582 + 1205852119896 + 1 + 120592 + 120582

2 100381610038161003816100381610038161003816100381610038161003816minus

11988811990924119896 ]]

]

(34)

Corollary 12 Let 120592 120582 120585 isin C be such that R(120592) gt 0 R(120582 +120585) gt 0 R(120585) gt minus1 and 120582119896 + 120585119896 = 0 minus1 sdot sdot sdot then thefollowing result holds

(119868(120592)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120582119896)+(120585119896)+120592minus1(2119896)(120585119896)

sdot Γ ((120582 + 120585) 119896)Γ (1 + 120585119896) Γ ((120582 + 120585) 119896 minus 120583)

times 21198653 [[[

120582 + 1205852119896 120582 + 120585

2119896 + 12 120585

119896 + 1 120582 + 1205852119896 minus 120583

2 120582 + 1205852119896 minus 120583 + 1

2 100381610038161003816100381610038161003816100381610038161003816minus

11988811990924119896 ]]

](35)

Corollary 13 Let 120592 120574 120585 120582 isin C be such thatR(120592) gt 0R(120582 +120585) gt 0R(120585) gt minus1 and let 120582119896 + 120585119896 + 120574 minus 120583 = 0 minus1 sdot sdot sdot thenthe following result holds

(119870+(120592120574)119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120585119896)+(120582119896)minus1(2119896)(120585119896)

sdot Γ ((120582 + 120585) 119896 + 120574)Γ (1 + 120585119896) Γ ((120582 + 120585) 119896 + 120592 + 120574)

times21198653 [[[

120582 + 1205852119896 + 120574

2 120582 + 1205852119896 + 120574 + 1

2120585119896 + 1 120582 + 120585

119896 + 120574 + 1205922 120582 + 120585

2119896 + 1 + 120592 + 1205742

100381610038161003816100381610038161003816100381610038161003816minus11988811990924119896 ]]

]

(36)

Theorem 14 Letting 120592 1205921015840 120583 1205831015840 120578 120582 V isin C be such that 119896 gt 0R(120574) gt 0R(]) gt minus1R(120582 minus V minus 2119899 minus 1) le 1 +R(120582 minus V minus 1) lt1+minR(minus120583)R(120592+ 1205921015840 minus120578)R(120592+ 1205831015840 minus120578) then there holdsthe following formula

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896V119888 (1119905 )) (119909) = 119909((120582minus])119896)+120578minus120592minus1205921015840minus1

(2119896)(]119896) times Γ (1 + (] minus 120582) 119896 minus 120583) Γ (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840) Γ (1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578)Γ (V119896 + 1) Γ (1 + (] minus 120582) 119896) Γ (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 + 120592 minus 120583)

times61198657 [[[[

1 + 120592 + 1205921015840 minus 1205782 + ] minus 120582

2119896 1 + ] minus 1205822119896 + 120592 + 1205921015840 minus 120578

2 1 + 120592 + 1205831015840 minus 1205782 + ] minus 120582

2119896 1 + ] minus 1205822119896 + 120592 + 1205831015840 minus 120578

2 1 minus 1205832 + ] minus 120582

2119896 1 + ] minus 1205822119896 minus 120583

2 V119896 + 1 1 + 120592 + 1205921015840 + 1205831015840 minus 120578

2 + ] minus 1205822119896 1 + ] minus 120582

2119896 + 120592 + 1205921015840 + 1205831015840 minus 1205782 12 + ] minus 120582

2119896 1 + ] minus 1205822119896 1 + 120592 minus 120583

2 + ] minus 1205822119896 1 + ] minus 120582

2119896 + 120592 minus 1205832

1003816100381610038161003816100381610038161003816minus119888

41198961199092]]]](37)

Proof Using (11) with 119911 = V119896 + 1 and (26) and applying (31)with 119911 being replaced by (]minus120582)119896minus120583+1 (]minus120582)119896+120592+1205831015840minus120578+1and (] minus 120582)119896 minus 120578 + 120592 + 1205921015840 + 1 we have

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896V119888 (1119905 )) (119909) = 119909((120582minus])119896)+120578minus120592minus1205921015840minus1

(2119896)(]119896)

times infinsum119899=0

Γ (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840) Γ (1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 minus 120583)Γ (1 + (] minus 120582) 119896) Γ (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 + 120592 minus 120583) Γ (V119896 + 1)

times (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840)2119899

(1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578)2119899

(1 + (] minus 120582) 119896 minus 120583)2119899(1 + (] minus 120582) 119896)2119899 (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578)2119899 (1 + (] minus 120582) 119896 + 120592 minus 120583)2119899 (V119896 + 1)119899

(minus119888)119899(41198961199092)119899 119899

= 119909((120582minus])119896)+120578minus120592minus1205921015840minus1(2119896)(]119896)

Γ (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840) Γ (1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 minus 120583)Γ (1 + (] minus 120582) 119896) Γ (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 + 120592 minus 120583) Γ (V119896 + 1)

times infinsum119899=0

(12 + (] minus 120582) 2119896 + (120592 + 1205921015840 minus 120578) 2)119899(1 + (] minus 120582) 2119896 + (120592 + 1205921015840 minus 120578) 2)

119899(V119896 + 1)119899 (12 + (] minus 120582) 2119896 + (120592 + 1205921015840 + 1205831015840 minus 120578) 2)119899 (1 + (] minus 120582) 2119896 + (120592 + 1205921015840 + 1205831015840 minus 120578) 2)119899

Journal of Mathematics 7

times (12 + (] minus 120582) 2119896 + (120592 + 1205831015840 minus 120578) 2)119899(1 + (] minus 120582) 2119896 + (120592 + 1205831015840 minus 120578) 2)

119899(12 + (] minus 120582) 2119896 minus 1205832)119899 (1 + (] minus 120582) 2119896 minus 1205832)119899

(12 + (] minus 120582) 2119896)119899 (1 + (] minus 120582) 2119896)119899 (1 + (] minus 120582) 2119896 + (120592 minus 120583) 2)119899 (12 + (] minus 120582) 2119896 + (120592 minus 120583) 2)119899sdot (minus119888)119899(41198961199092)119899 119899

(38)

Thus in accordance with (9) we get the required result (37)

Corollary 15 Let 120592 120583 120574 120585 120582 isin C and 119896 gt 0 be such thatR(120592) gt 0R(120585) gt minus1 R(V) gt minus1R(120592) gt 0R(120582 minus

V) lt 1 + min[R(120583)R(120574)] and let (120583 minus 120582)119896 + 120585119896 +1 120574 minus 120582119896 + 120585119896 + 1 = 0 minus1 sdot sdot sdot then the following resultholds

(119868(120592120583120574)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909)

= 119909(120582119896)minus(120585119896)minus120583minus1(2119896)(120585119896) times Γ ((120585 minus 120582) 119896 + 120574 + 1) Γ ((120585 minus 120582) 119896 + 120583 + 1)

Γ (120585119896 + 1) Γ (1 + (120585 minus 120582) 119896) Γ ((120585 minus 120582) 119896 + 120592 + 120583 + 120578 + 1)

times41198655 [[[

120583 + 12 + 120585 minus 120582

2119896 120583 + 22 + 120585 minus 120582

2119896 120574 + 12 + 120585 minus 120582

2119896 120574 + 22 + 120585 minus 120582

2119896 V119896 + 1 12 + 120585 minus 120582

2119896 1 + 120585 minus 1205822119896 120592 + 120583 + 120574 + 1

2 + 120585 minus 120582119896 120592 + 120574 + 2

2 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]]

(39)

Corollary 16 Let 120592 120574 120585 120582 isin C and 119896 gt 0 be such that 0 ltR(120592) lt 1minusR(120582minus120585)R(120585) gt minus1 and let120582119896minus120585119896+120592 = 1 2 sdot sdot sdot then the following result holds

(119868(120592)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120582119896)minus(120585119896)+120592minus1

(2119896)(120585119896)sdot Γ (1 minus 120592 + (120585 minus 120582) 119896)Γ (120585119896 + 1) Γ (1 + (120585 minus 120582) 119896)

times 21198653 [[[

1 minus 1205832 + 120585 minus 120582

2119896 2 minus 1205922 + 120585 minus 120582

2119896 120585119896 + 1 12 + 120585 minus 120582

2119896 1 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]](40)

Corollary 17 Let 120592 120574 120585 120582 isin C and 119896 gt 0 be such thatR(120585) gtminus1R(120592) gt 0R(120582+120585) lt 1+max[0R(120574)] and let 120582119896minus120585119896minus120574 = 1 2 sdot sdot sdot then the following formula holds

(119870minus(120592120574)119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120582119896)minus(120585119896)minus1

(2119896)(120585119896)Γ ((120585 minus 120582) 119896 + 120574 + 1)

Γ (1 + (120585 minus 120582) 119896) Γ (120585119896 + 1)

times 21198653 [[[

120574 + 12 + 120585 minus 120582

2119896 120574 + 22 + 120585 minus 120582

2119896 120585119896 + 1 120592 + 120574 + 1

2 + 120585 minus 120582119896 120592 + 120574 + 2

2 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]]

(41)

4 Concluding Remark

MSM fractional integral operators have advantage that theygeneralize theR-LWeyl Erdelyi-Kober and Saigorsquos fractionalintegral operators therefore several authors called this ageneral operator So we conclude this paper by emphasizingthat many other interesting image formulas can be derivedas the specific cases of our leading results Theorems 2 and6 involving familiar fractional integral operators as abovesaid Further the generalized Bessel function defined in (1)

possesses the lead that a number of Bessel functions trigono-metric functions and hyperbolic functions happen to bethe particular cases of this function Some special cases ofintegrals involving generalized Bessel function have beenexplored in the literature by a number of authors ([20ndash26])with different arguments Therefore results presented in thispaper are easily converted in terms of a comparable typeof novel interesting integrals with diverse arguments aftervarious suitable parametric replacements

8 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] S R Mondal Representation Formulae and Monotonicity ofthe Generalized k-Bessel Functions 2016

[2] Diaz and E Pariguan ldquoOn hypergeometric functions andPochhammer k-symbolrdquo Divulgaciones Matematicas vol 15no 2 pp 179ndash192 2007

[3] H SatohHigher Transcendental Functions vol 2McGraw-HillNewYork Toronto London 1953

[4] P AgarwalM Chand J Choi andG Singh ldquoCertain fractionalintegrals and image formulas of generalized k-Bessel functionrdquoCommunications of the KoreanMathematical Society vol 33 no2 pp 423ndash436 2018

[5] M Chand P Agarwal and Z Hammouch ldquoCertain SequencesInvolving Product of k-Bessel Functionrdquo International Journalof Applied and Computational Mathematics vol 4 no 4 4101pages 2018

[6] K S Gehlot ldquoDifferential Equation of K -Bessels Functions andits Propertiesrdquo Nonl Analysis and Differential Equations vol 2no 2 pp 61ndash67 2014

[7] K S Gehlot ldquoRecurrence Relations of K -Bessels functionrdquoThaiJ Math

[8] K S Gehlot and S D Purohit ldquoIntegral representations of thek-Besselrsquos functionrdquo Honam Mathematical Journal vol 38 no1 pp 17ndash23 2016

[9] G Singh P AgarwalM Chand and S Jain ldquo Certain fractionalkinetic equations involving generalized rdquo Transactions of ARazmadze Mathematical Institute 2018

[10] M Saigo and N Maeda ldquoMore generalization of fractionalcalculusrdquo in TransformMethods and Special Functions pp 386ndash400 Bulgarian Acad Sci Sofia 1998

[11] O I Marichev ldquoVolterra equation of Mellin convolution typewith a Horn function in the kernelrdquo Izvestiya Akademii NaukBSSR Seriya Fiziko-Matematicheskikh Nauk vol 1 pp 128-1291974 (Russian)

[12] H M Srivastava and P W Karlsson Multiple Gaussian hyper-geometric series Ellis Horwood Series Mathematics and itsApplications Ellis Horwood Ltd Chichester Halsted Press[John Wiley amp Sons Inc] New York NY USA 1985

[13] F W J Olver DW Lozier R F Boisvert and CW Clark EdsNIST Handbook of Mathematical Functions National Instituteof Standards and Technology USA Cambridge University PressGaithersburg Md 2010

[14] A Erdersquolyi W Magnus F Oberhettinger and F G Tri-comi Higher Transcendental Functions vol 1 McGraw-HillNewYork Toronto London 1953

[15] E M Wright ldquoThe asymptotic expansion of the generalizedhypergeometric functionrdquo Journal of the London MathematicalSociety vol 1-10 no 4 pp 286ndash293 1935

[16] E M Wright ldquoThe asymptotic expansion of the generalizedhypergeometric functionrdquo Proceedings of the London Mathe-matical Society vol 46 no 2 pp 389ndash408 1940

[17] E M Wright ldquoThe asymptotic expansion of integral functionsdefined by Taylor seriesrdquo Philosophical Transactions of the RoyalSociety A Mathematical Physical amp Engineering Sciences vol238 pp 423ndash451 1940

[18] A A Kilbas M Saigo and J J Trujillo ldquoOn the generalizedWright functionrdquo Fractional Calculus and Applied Analysis vol5 no 4 pp 437ndash460 2002

[19] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash1431978

[20] A A Kilbas and N Sebastian ldquoGeneralized fractional integra-tion of Bessel function of the first kindrdquo Integral Transforms andSpecial Functions vol 19 no 11-12 pp 869ndash883 2008

[21] H Amsalu and D L Suthar ldquoGeneralized Fractional IntegralOperators Involving Mittag-Leffler Functionrdquo Abstract andApplied Analysis vol 2018 Article ID 7034124 8 pages 2018

[22] S R Mondal and K S Nisar ldquoMarichev-Saigo-Maeda Frac-tional Integration Operators Involving Generalized BesselFunctionsrdquo Mathematical Problems in Engineering vol 2014Article ID 274093 11 pages 2014

[23] K S Nisar D L Suthar S D Purohit and M AldhaifallahldquoSome unified integrals associated with the generalized Struvefunctionrdquo Proceedings of the Jangjeon Mathematical SocietyMemoirs of the Jangjeon Mathematical Society vol 20 no 2 pp261ndash267 2017

[24] S D Purohit D L Suthar and S L Kalla ldquoMarichev-Saigo-Maeda fractional integration operators of the Bessel functionsrdquoLe Matematiche vol 67 no 1 pp 21ndash32 2012

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

[26] D L Suthar and H Habenom ldquoIntegrals involving generalizedBessel-Maitland functionrdquo Journal of Science andArts no 4(37)pp 357ndash362 2016

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4 Journal of Mathematics

= 119909((120582minus120585)119896)minus120592minus1205921015840+120578minus1(2119896)(120585119896)

times 3Ψ4 [[[[

(1 + 120585 minus 120582119896 minus 120583 2) (1 + 120585 minus 120582

119896 + 120592 + 1205921015840 minus 120578 2) (1 + 120585 minus 120582119896 + 120592 + 1205831015840 minus 120578 2)

(120585119896 + 1 1) (1 + 120585 minus 120582

119896 2) (1 + 120585 minus 120582119896 + 120592 + 1205921015840 + 1205831015840 minus 120578 2) (1 + 120585 minus 120582

119896 + 120592 minus 120583 2) 1003816100381610038161003816100381610038161003816minus

11988841198961199092

]]]]

(24)

Proof Using (2) and writing the function in the series formthe left-hand side of (24) leads to

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909)

= infinsum119899=0

(minus119888)119899 (12)(120585119896)+2119899Γ119896 (120585 + 119896 + 119899119896) 119899

times (119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus(120585119896)minus2119899minus1) (119909) (25)

Now upon using the image formula (17) which is validunder the conditions declared with Theorem 6 weget

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909)

= 119909(120582119896)minus(120585119896)minus120592minus1205921015840+120578minus1(2119896)(]119896)

times infinsum119899=0

Γ (1 + (120585 minus 120582) 119896 + 120592 + 1205921015840 minus 120578 + 2119899) Γ (1 + (120585 minus 120582) 119896 + 120592 + 1205831015840 minus 120578 + 2119899)Γ (1 + (120585 minus 120582) 119896 + 2119899) Γ (1 + (120585 minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578 + 2119899) Γ (1 + (120585 minus 120582) 119896 + 120592 minus 120583 + 2119899)

times Γ (1 + (120585 minus 120582) 119896 minus 120583 + 2119899)Γ (120585119896 + 1 + 119899)

(minus119888)119899(41198961199092)119899 119899

(26)

Using the definition of (15) in the right-hand side of (26) wearrive at the result (24)

Special Cases of Theorem 6(iv) If we substitute 1205921015840 = 0 120583 = minus120574 120578 = 120592 and replace120592 by 120592 + 120583 in (24) then we get the subsequent corollary

relating to right-hand sided Saigo fractional integral operator[19]

Corollary 7 Letting 120592 120583 120574 120582 120585 isin C andR(120585) gt minus1R(120592) gt0R(120582minus120585) lt 1+min[R(120583)R(120574)] then the following formulaholds

(119868(120592120583120574)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909((120582minus120585)119896)minus120583minus1

(2119896)(120585119896)

times 2Ψ3 [[[[

(1 + 120583 + 120585 minus 120582119896 2) (1 + 120585 minus 120582

119896 + 120574 2) (1 + 120585 minus 120582

119896 2) (1 + 120583 + 120592 + 120574 + 120585 minus 120582119896 2) (120585

119896 + 1 1) 1003816100381610038161003816100381610038161003816minus

11988841199092

]]]]

(27)

(v) If we set 120583 = minus120592 in (27) then we get the followingcorollary relating to right-sided Weyl fractional type integraloperator

Corollary 8 Let 120592 120582 120585 isin C be such that R(120592) gt 0R(120585) gt minus1 R(120592) lt 1 minus R(120582 minus 120585) then the following resultholds

(119868(120592)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120585119896)minus(120582119896)+120592minus1

(2119896)(120585119896)

times 1Ψ2 [[[[

(1 minus 120592 + 120585 minus 120582119896 2)

(120585 minus 120582119896 + 1 2) (120585

119896 + 1 1) 1003816100381610038161003816100381610038161003816minus

11988841198961199092

]]]](28)

Journal of Mathematics 5

(vi) If we set 120583 = 0 in (27) then we get the subsequentcorollary relating to right-hand side of Erdelyi-Kober frac-tional type integral operator

Corollary 9 Let 120592 120574120582 120585 isin C be such thatR(120585) gt minus1R(120592) gt0 R(120582 + 120585) lt 1 + min[0R(120574)] then the following formulaholds

(119870minus(120592120574)119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120585119896)minus(120582119896)minus1

(2119896)(120585119896)

times 1Ψ2 [[[[

(1 + 120585 minus 120582119896 + 120574 2)

(1 + 120585 minus 120582119896 + 120592 + 120574 2) ( 120585

119896 + 1 1) 1003816100381610038161003816100381610038161003816minus

11988841198961199092

]]]](29)

3 Representation in Terms of GeneralizedHypergeometric Function

In this part we introduce the generalized fractionalintegrals of 119896-Bessel function in terms of generalized

hypergeometric function We consider the followingwell-known results

Γ (2119911) = 22119911minus1radic120587 Γ (119911) Γ (119911 + 1

2) 119911 isin C (30)

and

(119911)2119899 = 22119899 (1199112)119899 (

1199112 + 1

2)119899 119911 isin C 119899 isin N (31)

We represent the following theorems containing the general-ized hypergeometric function

Theorem 10 Let 120592 1205921015840 120583 1205831015840 120578 120582 120585 isin C be such thatR(120578) gt 0R(120585) gt minus1 R(120582 + 120585 + 2119899119896) ge R(120582 + 120585) gt max[0R(120592 +1205921015840 + 120583 minus 120578)R(1205921015840 minus 1205831015840)] and let (120582 + 120585)119896 (120582 + 120585)119896 minus120592 minus 120592 minus 120583 + 120578 = 0 minus1 sdot sdot sdot then the following formulaholds

(119868(12059212059210158401205831205831015840120578)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909((120582+120585)119896)minus120592minus1205921015840+120578minus1(2119896)(120585119896) times Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578) Γ ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)

Γ ((120582 + 120585) 119896 + 1205831015840) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120592 + 120578) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120583 + 120578) Γ (120585119896 + 1)

times 61198657 [[[[

120582 + 1205852119896 120582 + 120585

2119896 + 12

120582 + 1205852119896 + 120578 minus 120592 minus 120592 minus 120583

2 120582 + 1205852119896 + 120578 minus 120592 minus 120592 minus 120583 + 1

2 120582 + 1205852119896 + 1205831015840 minus 1205921015840

2 120582 + 1205852119896 + 1205831015840 minus 1205921015840 + 1

2 120585119896 + 1 120582 + 120585

2119896 + 12058310158402 120582 + 120585

2119896 + 1205831015840 + 12 120582 + 120585

2119896 + 120578 minus 120592 minus 12059210158402 120582 + 120585

2119896 + 120578 minus 120592 minus 1205921015840 + 12 120582 + 120585

2119896 + 120578 minus 1205921015840 minus 1205832 120582 + 120585

2119896 + 120578 minus 1205921015840 minus 120583 + 12

100381610038161003816100381610038161003816100381610038161003816minus11988811990924119896

]]]](32)

Proof Note that 61198657 defined in (32) exit as the series isabsolutely convergent Now using (11) with 119911 = 120585119896 + 1 and

(20) and applying (31) with 119911 being replaced by (120582 + 120585)119896(120582 + 120585)119896 minus 120592 minus 120592 minus 120583 + 120578 and (120582 + 120585)119896 + 1205831015840 minus 1205921015840 we have

(119868(12059212059210158401205831205831015840120578)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909((120582+120585)119896)minus120592minus1205921015840+120578minus1(2119896)(120585119896)

times infinsum119899=0

Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578) Γ ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)Γ ((120582 + 120585) 119896 + 1205831015840) Γ ((120582 + 120585) 119896 minus 120592 minus 1205921015840 + 120578) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120583 + 120578) Γ (120585119896 + 1)

times ((120582 + 120585) 119896)2119899 ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578)2119899 ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)2119899((120582 + 120585) 119896 + 1205831015840)2119899 ((120582 + 120585) 119896 minus 120592 minus 1205921015840 + 120578)2119899 ((120582 + 120585) 119896 + 120578 minus 1205921015840 minus 120583)2119899 (120585119896 + 1)119899

(minus1198881199092)119899(4119896)119899 119899

= 119909((120582+120585)119896)minus120592minus1205921015840+120578minus1(2119896)(120585119896)

Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578) Γ ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)Γ ((120582 + 120585) 119896 + 1205831015840) Γ ((120582 + 120585) 119896 minus 120592 minus 1205921015840 + 120578) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120583 + 120578) Γ (120585119896 + 1)

times infinsum119899=0

((120582 + 120585) 2119896)119899 ((120582 + 120585) 2119896 + 12)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 120592 minus 120583) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 120592 minus 120583 + 1) 2)119899(120585119896 + 1)119899 ((120582 + 120585) 2119896 + 12058310158402)119899 ((120582 + 120585) 2119896 + (1205831015840 + 1) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 1205921015840) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 1205921015840 + 1) 2)119899

times ((120582 + 120585) 2119896 + (1205831015840 minus 1205921015840) 2)119899((120582 + 120585) 2119896 + (1205831015840 minus 1205921015840 + 1) 2)

119899((120582 + 120585) 2119896 + (120578 minus 1205921015840 minus 120583) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 1205921015840 minus 120583 + 1) 2)119899(minus1198881199092)119899(4119896)119899 119899

(33)

Thus in accordance with (9) we get the required result(32)

Corollary 11 Let 120592 120583 120574 120582 120585 isin C be such that R(120592) gt 0R(120585) gt minus1R(120582 + 120585) gt max[0R(120583 minus 120574)] and let (120582 + 120585)119896

6 Journal of Mathematics

(120582 + 120585)119896 + 120574 minus 120583 = 0 minus1 sdot sdot sdot then the following resultholds

(119868(120592120583120574)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120585119896)+(120582119896)minus120583minus1(2119896)(120585119896)

Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 + 120574 minus 120583)Γ (V119896 + 1) Γ ((120582 + 120585) 119896 minus 120583) Γ ((120582 + 120585) 119896 + 120592 + 120574)

times41198655 [[[

120582 + 1205852119896 120582 + 120585

2119896 + 12

120582 + 1205852119896 + 120574 minus 120583

2 120582 + 1205852119896 + 1 + 120574 minus 120583

2 V119896 + 1 120582 + 120585

2119896 minus 1205832

120582 + 1205852119896 minus 120583 + 1

2 120582 + 120585119896 + 120592 + 120574

2 120582 + 1205852119896 + 1 + 120592 + 120582

2 100381610038161003816100381610038161003816100381610038161003816minus

11988811990924119896 ]]

]

(34)

Corollary 12 Let 120592 120582 120585 isin C be such that R(120592) gt 0 R(120582 +120585) gt 0 R(120585) gt minus1 and 120582119896 + 120585119896 = 0 minus1 sdot sdot sdot then thefollowing result holds

(119868(120592)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120582119896)+(120585119896)+120592minus1(2119896)(120585119896)

sdot Γ ((120582 + 120585) 119896)Γ (1 + 120585119896) Γ ((120582 + 120585) 119896 minus 120583)

times 21198653 [[[

120582 + 1205852119896 120582 + 120585

2119896 + 12 120585

119896 + 1 120582 + 1205852119896 minus 120583

2 120582 + 1205852119896 minus 120583 + 1

2 100381610038161003816100381610038161003816100381610038161003816minus

11988811990924119896 ]]

](35)

Corollary 13 Let 120592 120574 120585 120582 isin C be such thatR(120592) gt 0R(120582 +120585) gt 0R(120585) gt minus1 and let 120582119896 + 120585119896 + 120574 minus 120583 = 0 minus1 sdot sdot sdot thenthe following result holds

(119870+(120592120574)119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120585119896)+(120582119896)minus1(2119896)(120585119896)

sdot Γ ((120582 + 120585) 119896 + 120574)Γ (1 + 120585119896) Γ ((120582 + 120585) 119896 + 120592 + 120574)

times21198653 [[[

120582 + 1205852119896 + 120574

2 120582 + 1205852119896 + 120574 + 1

2120585119896 + 1 120582 + 120585

119896 + 120574 + 1205922 120582 + 120585

2119896 + 1 + 120592 + 1205742

100381610038161003816100381610038161003816100381610038161003816minus11988811990924119896 ]]

]

(36)

Theorem 14 Letting 120592 1205921015840 120583 1205831015840 120578 120582 V isin C be such that 119896 gt 0R(120574) gt 0R(]) gt minus1R(120582 minus V minus 2119899 minus 1) le 1 +R(120582 minus V minus 1) lt1+minR(minus120583)R(120592+ 1205921015840 minus120578)R(120592+ 1205831015840 minus120578) then there holdsthe following formula

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896V119888 (1119905 )) (119909) = 119909((120582minus])119896)+120578minus120592minus1205921015840minus1

(2119896)(]119896) times Γ (1 + (] minus 120582) 119896 minus 120583) Γ (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840) Γ (1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578)Γ (V119896 + 1) Γ (1 + (] minus 120582) 119896) Γ (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 + 120592 minus 120583)

times61198657 [[[[

1 + 120592 + 1205921015840 minus 1205782 + ] minus 120582

2119896 1 + ] minus 1205822119896 + 120592 + 1205921015840 minus 120578

2 1 + 120592 + 1205831015840 minus 1205782 + ] minus 120582

2119896 1 + ] minus 1205822119896 + 120592 + 1205831015840 minus 120578

2 1 minus 1205832 + ] minus 120582

2119896 1 + ] minus 1205822119896 minus 120583

2 V119896 + 1 1 + 120592 + 1205921015840 + 1205831015840 minus 120578

2 + ] minus 1205822119896 1 + ] minus 120582

2119896 + 120592 + 1205921015840 + 1205831015840 minus 1205782 12 + ] minus 120582

2119896 1 + ] minus 1205822119896 1 + 120592 minus 120583

2 + ] minus 1205822119896 1 + ] minus 120582

2119896 + 120592 minus 1205832

1003816100381610038161003816100381610038161003816minus119888

41198961199092]]]](37)

Proof Using (11) with 119911 = V119896 + 1 and (26) and applying (31)with 119911 being replaced by (]minus120582)119896minus120583+1 (]minus120582)119896+120592+1205831015840minus120578+1and (] minus 120582)119896 minus 120578 + 120592 + 1205921015840 + 1 we have

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896V119888 (1119905 )) (119909) = 119909((120582minus])119896)+120578minus120592minus1205921015840minus1

(2119896)(]119896)

times infinsum119899=0

Γ (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840) Γ (1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 minus 120583)Γ (1 + (] minus 120582) 119896) Γ (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 + 120592 minus 120583) Γ (V119896 + 1)

times (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840)2119899

(1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578)2119899

(1 + (] minus 120582) 119896 minus 120583)2119899(1 + (] minus 120582) 119896)2119899 (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578)2119899 (1 + (] minus 120582) 119896 + 120592 minus 120583)2119899 (V119896 + 1)119899

(minus119888)119899(41198961199092)119899 119899

= 119909((120582minus])119896)+120578minus120592minus1205921015840minus1(2119896)(]119896)

Γ (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840) Γ (1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 minus 120583)Γ (1 + (] minus 120582) 119896) Γ (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 + 120592 minus 120583) Γ (V119896 + 1)

times infinsum119899=0

(12 + (] minus 120582) 2119896 + (120592 + 1205921015840 minus 120578) 2)119899(1 + (] minus 120582) 2119896 + (120592 + 1205921015840 minus 120578) 2)

119899(V119896 + 1)119899 (12 + (] minus 120582) 2119896 + (120592 + 1205921015840 + 1205831015840 minus 120578) 2)119899 (1 + (] minus 120582) 2119896 + (120592 + 1205921015840 + 1205831015840 minus 120578) 2)119899

Journal of Mathematics 7

times (12 + (] minus 120582) 2119896 + (120592 + 1205831015840 minus 120578) 2)119899(1 + (] minus 120582) 2119896 + (120592 + 1205831015840 minus 120578) 2)

119899(12 + (] minus 120582) 2119896 minus 1205832)119899 (1 + (] minus 120582) 2119896 minus 1205832)119899

(12 + (] minus 120582) 2119896)119899 (1 + (] minus 120582) 2119896)119899 (1 + (] minus 120582) 2119896 + (120592 minus 120583) 2)119899 (12 + (] minus 120582) 2119896 + (120592 minus 120583) 2)119899sdot (minus119888)119899(41198961199092)119899 119899

(38)

Thus in accordance with (9) we get the required result (37)

Corollary 15 Let 120592 120583 120574 120585 120582 isin C and 119896 gt 0 be such thatR(120592) gt 0R(120585) gt minus1 R(V) gt minus1R(120592) gt 0R(120582 minus

V) lt 1 + min[R(120583)R(120574)] and let (120583 minus 120582)119896 + 120585119896 +1 120574 minus 120582119896 + 120585119896 + 1 = 0 minus1 sdot sdot sdot then the following resultholds

(119868(120592120583120574)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909)

= 119909(120582119896)minus(120585119896)minus120583minus1(2119896)(120585119896) times Γ ((120585 minus 120582) 119896 + 120574 + 1) Γ ((120585 minus 120582) 119896 + 120583 + 1)

Γ (120585119896 + 1) Γ (1 + (120585 minus 120582) 119896) Γ ((120585 minus 120582) 119896 + 120592 + 120583 + 120578 + 1)

times41198655 [[[

120583 + 12 + 120585 minus 120582

2119896 120583 + 22 + 120585 minus 120582

2119896 120574 + 12 + 120585 minus 120582

2119896 120574 + 22 + 120585 minus 120582

2119896 V119896 + 1 12 + 120585 minus 120582

2119896 1 + 120585 minus 1205822119896 120592 + 120583 + 120574 + 1

2 + 120585 minus 120582119896 120592 + 120574 + 2

2 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]]

(39)

Corollary 16 Let 120592 120574 120585 120582 isin C and 119896 gt 0 be such that 0 ltR(120592) lt 1minusR(120582minus120585)R(120585) gt minus1 and let120582119896minus120585119896+120592 = 1 2 sdot sdot sdot then the following result holds

(119868(120592)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120582119896)minus(120585119896)+120592minus1

(2119896)(120585119896)sdot Γ (1 minus 120592 + (120585 minus 120582) 119896)Γ (120585119896 + 1) Γ (1 + (120585 minus 120582) 119896)

times 21198653 [[[

1 minus 1205832 + 120585 minus 120582

2119896 2 minus 1205922 + 120585 minus 120582

2119896 120585119896 + 1 12 + 120585 minus 120582

2119896 1 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]](40)

Corollary 17 Let 120592 120574 120585 120582 isin C and 119896 gt 0 be such thatR(120585) gtminus1R(120592) gt 0R(120582+120585) lt 1+max[0R(120574)] and let 120582119896minus120585119896minus120574 = 1 2 sdot sdot sdot then the following formula holds

(119870minus(120592120574)119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120582119896)minus(120585119896)minus1

(2119896)(120585119896)Γ ((120585 minus 120582) 119896 + 120574 + 1)

Γ (1 + (120585 minus 120582) 119896) Γ (120585119896 + 1)

times 21198653 [[[

120574 + 12 + 120585 minus 120582

2119896 120574 + 22 + 120585 minus 120582

2119896 120585119896 + 1 120592 + 120574 + 1

2 + 120585 minus 120582119896 120592 + 120574 + 2

2 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]]

(41)

4 Concluding Remark

MSM fractional integral operators have advantage that theygeneralize theR-LWeyl Erdelyi-Kober and Saigorsquos fractionalintegral operators therefore several authors called this ageneral operator So we conclude this paper by emphasizingthat many other interesting image formulas can be derivedas the specific cases of our leading results Theorems 2 and6 involving familiar fractional integral operators as abovesaid Further the generalized Bessel function defined in (1)

possesses the lead that a number of Bessel functions trigono-metric functions and hyperbolic functions happen to bethe particular cases of this function Some special cases ofintegrals involving generalized Bessel function have beenexplored in the literature by a number of authors ([20ndash26])with different arguments Therefore results presented in thispaper are easily converted in terms of a comparable typeof novel interesting integrals with diverse arguments aftervarious suitable parametric replacements

8 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] S R Mondal Representation Formulae and Monotonicity ofthe Generalized k-Bessel Functions 2016

[2] Diaz and E Pariguan ldquoOn hypergeometric functions andPochhammer k-symbolrdquo Divulgaciones Matematicas vol 15no 2 pp 179ndash192 2007

[3] H SatohHigher Transcendental Functions vol 2McGraw-HillNewYork Toronto London 1953

[4] P AgarwalM Chand J Choi andG Singh ldquoCertain fractionalintegrals and image formulas of generalized k-Bessel functionrdquoCommunications of the KoreanMathematical Society vol 33 no2 pp 423ndash436 2018

[5] M Chand P Agarwal and Z Hammouch ldquoCertain SequencesInvolving Product of k-Bessel Functionrdquo International Journalof Applied and Computational Mathematics vol 4 no 4 4101pages 2018

[6] K S Gehlot ldquoDifferential Equation of K -Bessels Functions andits Propertiesrdquo Nonl Analysis and Differential Equations vol 2no 2 pp 61ndash67 2014

[7] K S Gehlot ldquoRecurrence Relations of K -Bessels functionrdquoThaiJ Math

[8] K S Gehlot and S D Purohit ldquoIntegral representations of thek-Besselrsquos functionrdquo Honam Mathematical Journal vol 38 no1 pp 17ndash23 2016

[9] G Singh P AgarwalM Chand and S Jain ldquo Certain fractionalkinetic equations involving generalized rdquo Transactions of ARazmadze Mathematical Institute 2018

[10] M Saigo and N Maeda ldquoMore generalization of fractionalcalculusrdquo in TransformMethods and Special Functions pp 386ndash400 Bulgarian Acad Sci Sofia 1998

[11] O I Marichev ldquoVolterra equation of Mellin convolution typewith a Horn function in the kernelrdquo Izvestiya Akademii NaukBSSR Seriya Fiziko-Matematicheskikh Nauk vol 1 pp 128-1291974 (Russian)

[12] H M Srivastava and P W Karlsson Multiple Gaussian hyper-geometric series Ellis Horwood Series Mathematics and itsApplications Ellis Horwood Ltd Chichester Halsted Press[John Wiley amp Sons Inc] New York NY USA 1985

[13] F W J Olver DW Lozier R F Boisvert and CW Clark EdsNIST Handbook of Mathematical Functions National Instituteof Standards and Technology USA Cambridge University PressGaithersburg Md 2010

[14] A Erdersquolyi W Magnus F Oberhettinger and F G Tri-comi Higher Transcendental Functions vol 1 McGraw-HillNewYork Toronto London 1953

[15] E M Wright ldquoThe asymptotic expansion of the generalizedhypergeometric functionrdquo Journal of the London MathematicalSociety vol 1-10 no 4 pp 286ndash293 1935

[16] E M Wright ldquoThe asymptotic expansion of the generalizedhypergeometric functionrdquo Proceedings of the London Mathe-matical Society vol 46 no 2 pp 389ndash408 1940

[17] E M Wright ldquoThe asymptotic expansion of integral functionsdefined by Taylor seriesrdquo Philosophical Transactions of the RoyalSociety A Mathematical Physical amp Engineering Sciences vol238 pp 423ndash451 1940

[18] A A Kilbas M Saigo and J J Trujillo ldquoOn the generalizedWright functionrdquo Fractional Calculus and Applied Analysis vol5 no 4 pp 437ndash460 2002

[19] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash1431978

[20] A A Kilbas and N Sebastian ldquoGeneralized fractional integra-tion of Bessel function of the first kindrdquo Integral Transforms andSpecial Functions vol 19 no 11-12 pp 869ndash883 2008

[21] H Amsalu and D L Suthar ldquoGeneralized Fractional IntegralOperators Involving Mittag-Leffler Functionrdquo Abstract andApplied Analysis vol 2018 Article ID 7034124 8 pages 2018

[22] S R Mondal and K S Nisar ldquoMarichev-Saigo-Maeda Frac-tional Integration Operators Involving Generalized BesselFunctionsrdquo Mathematical Problems in Engineering vol 2014Article ID 274093 11 pages 2014

[23] K S Nisar D L Suthar S D Purohit and M AldhaifallahldquoSome unified integrals associated with the generalized Struvefunctionrdquo Proceedings of the Jangjeon Mathematical SocietyMemoirs of the Jangjeon Mathematical Society vol 20 no 2 pp261ndash267 2017

[24] S D Purohit D L Suthar and S L Kalla ldquoMarichev-Saigo-Maeda fractional integration operators of the Bessel functionsrdquoLe Matematiche vol 67 no 1 pp 21ndash32 2012

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

[26] D L Suthar and H Habenom ldquoIntegrals involving generalizedBessel-Maitland functionrdquo Journal of Science andArts no 4(37)pp 357ndash362 2016

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Journal of Mathematics 5

(vi) If we set 120583 = 0 in (27) then we get the subsequentcorollary relating to right-hand side of Erdelyi-Kober frac-tional type integral operator

Corollary 9 Let 120592 120574120582 120585 isin C be such thatR(120585) gt minus1R(120592) gt0 R(120582 + 120585) lt 1 + min[0R(120574)] then the following formulaholds

(119870minus(120592120574)119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120585119896)minus(120582119896)minus1

(2119896)(120585119896)

times 1Ψ2 [[[[

(1 + 120585 minus 120582119896 + 120574 2)

(1 + 120585 minus 120582119896 + 120592 + 120574 2) ( 120585

119896 + 1 1) 1003816100381610038161003816100381610038161003816minus

11988841198961199092

]]]](29)

3 Representation in Terms of GeneralizedHypergeometric Function

In this part we introduce the generalized fractionalintegrals of 119896-Bessel function in terms of generalized

hypergeometric function We consider the followingwell-known results

Γ (2119911) = 22119911minus1radic120587 Γ (119911) Γ (119911 + 1

2) 119911 isin C (30)

and

(119911)2119899 = 22119899 (1199112)119899 (

1199112 + 1

2)119899 119911 isin C 119899 isin N (31)

We represent the following theorems containing the general-ized hypergeometric function

Theorem 10 Let 120592 1205921015840 120583 1205831015840 120578 120582 120585 isin C be such thatR(120578) gt 0R(120585) gt minus1 R(120582 + 120585 + 2119899119896) ge R(120582 + 120585) gt max[0R(120592 +1205921015840 + 120583 minus 120578)R(1205921015840 minus 1205831015840)] and let (120582 + 120585)119896 (120582 + 120585)119896 minus120592 minus 120592 minus 120583 + 120578 = 0 minus1 sdot sdot sdot then the following formulaholds

(119868(12059212059210158401205831205831015840120578)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909((120582+120585)119896)minus120592minus1205921015840+120578minus1(2119896)(120585119896) times Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578) Γ ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)

Γ ((120582 + 120585) 119896 + 1205831015840) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120592 + 120578) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120583 + 120578) Γ (120585119896 + 1)

times 61198657 [[[[

120582 + 1205852119896 120582 + 120585

2119896 + 12

120582 + 1205852119896 + 120578 minus 120592 minus 120592 minus 120583

2 120582 + 1205852119896 + 120578 minus 120592 minus 120592 minus 120583 + 1

2 120582 + 1205852119896 + 1205831015840 minus 1205921015840

2 120582 + 1205852119896 + 1205831015840 minus 1205921015840 + 1

2 120585119896 + 1 120582 + 120585

2119896 + 12058310158402 120582 + 120585

2119896 + 1205831015840 + 12 120582 + 120585

2119896 + 120578 minus 120592 minus 12059210158402 120582 + 120585

2119896 + 120578 minus 120592 minus 1205921015840 + 12 120582 + 120585

2119896 + 120578 minus 1205921015840 minus 1205832 120582 + 120585

2119896 + 120578 minus 1205921015840 minus 120583 + 12

100381610038161003816100381610038161003816100381610038161003816minus11988811990924119896

]]]](32)

Proof Note that 61198657 defined in (32) exit as the series isabsolutely convergent Now using (11) with 119911 = 120585119896 + 1 and

(20) and applying (31) with 119911 being replaced by (120582 + 120585)119896(120582 + 120585)119896 minus 120592 minus 120592 minus 120583 + 120578 and (120582 + 120585)119896 + 1205831015840 minus 1205921015840 we have

(119868(12059212059210158401205831205831015840120578)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909((120582+120585)119896)minus120592minus1205921015840+120578minus1(2119896)(120585119896)

times infinsum119899=0

Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578) Γ ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)Γ ((120582 + 120585) 119896 + 1205831015840) Γ ((120582 + 120585) 119896 minus 120592 minus 1205921015840 + 120578) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120583 + 120578) Γ (120585119896 + 1)

times ((120582 + 120585) 119896)2119899 ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578)2119899 ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)2119899((120582 + 120585) 119896 + 1205831015840)2119899 ((120582 + 120585) 119896 minus 120592 minus 1205921015840 + 120578)2119899 ((120582 + 120585) 119896 + 120578 minus 1205921015840 minus 120583)2119899 (120585119896 + 1)119899

(minus1198881199092)119899(4119896)119899 119899

= 119909((120582+120585)119896)minus120592minus1205921015840+120578minus1(2119896)(120585119896)

Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 minus 120592 minus 120592 minus 120583 + 120578) Γ ((120582 + 120585) 119896 + 1205831015840 minus 1205921015840)Γ ((120582 + 120585) 119896 + 1205831015840) Γ ((120582 + 120585) 119896 minus 120592 minus 1205921015840 + 120578) Γ ((120582 + 120585) 119896 minus 1205921015840 minus 120583 + 120578) Γ (120585119896 + 1)

times infinsum119899=0

((120582 + 120585) 2119896)119899 ((120582 + 120585) 2119896 + 12)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 120592 minus 120583) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 120592 minus 120583 + 1) 2)119899(120585119896 + 1)119899 ((120582 + 120585) 2119896 + 12058310158402)119899 ((120582 + 120585) 2119896 + (1205831015840 + 1) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 1205921015840) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 120592 minus 1205921015840 + 1) 2)119899

times ((120582 + 120585) 2119896 + (1205831015840 minus 1205921015840) 2)119899((120582 + 120585) 2119896 + (1205831015840 minus 1205921015840 + 1) 2)

119899((120582 + 120585) 2119896 + (120578 minus 1205921015840 minus 120583) 2)119899 ((120582 + 120585) 2119896 + (120578 minus 1205921015840 minus 120583 + 1) 2)119899(minus1198881199092)119899(4119896)119899 119899

(33)

Thus in accordance with (9) we get the required result(32)

Corollary 11 Let 120592 120583 120574 120582 120585 isin C be such that R(120592) gt 0R(120585) gt minus1R(120582 + 120585) gt max[0R(120583 minus 120574)] and let (120582 + 120585)119896

6 Journal of Mathematics

(120582 + 120585)119896 + 120574 minus 120583 = 0 minus1 sdot sdot sdot then the following resultholds

(119868(120592120583120574)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120585119896)+(120582119896)minus120583minus1(2119896)(120585119896)

Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 + 120574 minus 120583)Γ (V119896 + 1) Γ ((120582 + 120585) 119896 minus 120583) Γ ((120582 + 120585) 119896 + 120592 + 120574)

times41198655 [[[

120582 + 1205852119896 120582 + 120585

2119896 + 12

120582 + 1205852119896 + 120574 minus 120583

2 120582 + 1205852119896 + 1 + 120574 minus 120583

2 V119896 + 1 120582 + 120585

2119896 minus 1205832

120582 + 1205852119896 minus 120583 + 1

2 120582 + 120585119896 + 120592 + 120574

2 120582 + 1205852119896 + 1 + 120592 + 120582

2 100381610038161003816100381610038161003816100381610038161003816minus

11988811990924119896 ]]

]

(34)

Corollary 12 Let 120592 120582 120585 isin C be such that R(120592) gt 0 R(120582 +120585) gt 0 R(120585) gt minus1 and 120582119896 + 120585119896 = 0 minus1 sdot sdot sdot then thefollowing result holds

(119868(120592)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120582119896)+(120585119896)+120592minus1(2119896)(120585119896)

sdot Γ ((120582 + 120585) 119896)Γ (1 + 120585119896) Γ ((120582 + 120585) 119896 minus 120583)

times 21198653 [[[

120582 + 1205852119896 120582 + 120585

2119896 + 12 120585

119896 + 1 120582 + 1205852119896 minus 120583

2 120582 + 1205852119896 minus 120583 + 1

2 100381610038161003816100381610038161003816100381610038161003816minus

11988811990924119896 ]]

](35)

Corollary 13 Let 120592 120574 120585 120582 isin C be such thatR(120592) gt 0R(120582 +120585) gt 0R(120585) gt minus1 and let 120582119896 + 120585119896 + 120574 minus 120583 = 0 minus1 sdot sdot sdot thenthe following result holds

(119870+(120592120574)119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120585119896)+(120582119896)minus1(2119896)(120585119896)

sdot Γ ((120582 + 120585) 119896 + 120574)Γ (1 + 120585119896) Γ ((120582 + 120585) 119896 + 120592 + 120574)

times21198653 [[[

120582 + 1205852119896 + 120574

2 120582 + 1205852119896 + 120574 + 1

2120585119896 + 1 120582 + 120585

119896 + 120574 + 1205922 120582 + 120585

2119896 + 1 + 120592 + 1205742

100381610038161003816100381610038161003816100381610038161003816minus11988811990924119896 ]]

]

(36)

Theorem 14 Letting 120592 1205921015840 120583 1205831015840 120578 120582 V isin C be such that 119896 gt 0R(120574) gt 0R(]) gt minus1R(120582 minus V minus 2119899 minus 1) le 1 +R(120582 minus V minus 1) lt1+minR(minus120583)R(120592+ 1205921015840 minus120578)R(120592+ 1205831015840 minus120578) then there holdsthe following formula

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896V119888 (1119905 )) (119909) = 119909((120582minus])119896)+120578minus120592minus1205921015840minus1

(2119896)(]119896) times Γ (1 + (] minus 120582) 119896 minus 120583) Γ (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840) Γ (1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578)Γ (V119896 + 1) Γ (1 + (] minus 120582) 119896) Γ (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 + 120592 minus 120583)

times61198657 [[[[

1 + 120592 + 1205921015840 minus 1205782 + ] minus 120582

2119896 1 + ] minus 1205822119896 + 120592 + 1205921015840 minus 120578

2 1 + 120592 + 1205831015840 minus 1205782 + ] minus 120582

2119896 1 + ] minus 1205822119896 + 120592 + 1205831015840 minus 120578

2 1 minus 1205832 + ] minus 120582

2119896 1 + ] minus 1205822119896 minus 120583

2 V119896 + 1 1 + 120592 + 1205921015840 + 1205831015840 minus 120578

2 + ] minus 1205822119896 1 + ] minus 120582

2119896 + 120592 + 1205921015840 + 1205831015840 minus 1205782 12 + ] minus 120582

2119896 1 + ] minus 1205822119896 1 + 120592 minus 120583

2 + ] minus 1205822119896 1 + ] minus 120582

2119896 + 120592 minus 1205832

1003816100381610038161003816100381610038161003816minus119888

41198961199092]]]](37)

Proof Using (11) with 119911 = V119896 + 1 and (26) and applying (31)with 119911 being replaced by (]minus120582)119896minus120583+1 (]minus120582)119896+120592+1205831015840minus120578+1and (] minus 120582)119896 minus 120578 + 120592 + 1205921015840 + 1 we have

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896V119888 (1119905 )) (119909) = 119909((120582minus])119896)+120578minus120592minus1205921015840minus1

(2119896)(]119896)

times infinsum119899=0

Γ (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840) Γ (1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 minus 120583)Γ (1 + (] minus 120582) 119896) Γ (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 + 120592 minus 120583) Γ (V119896 + 1)

times (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840)2119899

(1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578)2119899

(1 + (] minus 120582) 119896 minus 120583)2119899(1 + (] minus 120582) 119896)2119899 (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578)2119899 (1 + (] minus 120582) 119896 + 120592 minus 120583)2119899 (V119896 + 1)119899

(minus119888)119899(41198961199092)119899 119899

= 119909((120582minus])119896)+120578minus120592minus1205921015840minus1(2119896)(]119896)

Γ (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840) Γ (1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 minus 120583)Γ (1 + (] minus 120582) 119896) Γ (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 + 120592 minus 120583) Γ (V119896 + 1)

times infinsum119899=0

(12 + (] minus 120582) 2119896 + (120592 + 1205921015840 minus 120578) 2)119899(1 + (] minus 120582) 2119896 + (120592 + 1205921015840 minus 120578) 2)

119899(V119896 + 1)119899 (12 + (] minus 120582) 2119896 + (120592 + 1205921015840 + 1205831015840 minus 120578) 2)119899 (1 + (] minus 120582) 2119896 + (120592 + 1205921015840 + 1205831015840 minus 120578) 2)119899

Journal of Mathematics 7

times (12 + (] minus 120582) 2119896 + (120592 + 1205831015840 minus 120578) 2)119899(1 + (] minus 120582) 2119896 + (120592 + 1205831015840 minus 120578) 2)

119899(12 + (] minus 120582) 2119896 minus 1205832)119899 (1 + (] minus 120582) 2119896 minus 1205832)119899

(12 + (] minus 120582) 2119896)119899 (1 + (] minus 120582) 2119896)119899 (1 + (] minus 120582) 2119896 + (120592 minus 120583) 2)119899 (12 + (] minus 120582) 2119896 + (120592 minus 120583) 2)119899sdot (minus119888)119899(41198961199092)119899 119899

(38)

Thus in accordance with (9) we get the required result (37)

Corollary 15 Let 120592 120583 120574 120585 120582 isin C and 119896 gt 0 be such thatR(120592) gt 0R(120585) gt minus1 R(V) gt minus1R(120592) gt 0R(120582 minus

V) lt 1 + min[R(120583)R(120574)] and let (120583 minus 120582)119896 + 120585119896 +1 120574 minus 120582119896 + 120585119896 + 1 = 0 minus1 sdot sdot sdot then the following resultholds

(119868(120592120583120574)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909)

= 119909(120582119896)minus(120585119896)minus120583minus1(2119896)(120585119896) times Γ ((120585 minus 120582) 119896 + 120574 + 1) Γ ((120585 minus 120582) 119896 + 120583 + 1)

Γ (120585119896 + 1) Γ (1 + (120585 minus 120582) 119896) Γ ((120585 minus 120582) 119896 + 120592 + 120583 + 120578 + 1)

times41198655 [[[

120583 + 12 + 120585 minus 120582

2119896 120583 + 22 + 120585 minus 120582

2119896 120574 + 12 + 120585 minus 120582

2119896 120574 + 22 + 120585 minus 120582

2119896 V119896 + 1 12 + 120585 minus 120582

2119896 1 + 120585 minus 1205822119896 120592 + 120583 + 120574 + 1

2 + 120585 minus 120582119896 120592 + 120574 + 2

2 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]]

(39)

Corollary 16 Let 120592 120574 120585 120582 isin C and 119896 gt 0 be such that 0 ltR(120592) lt 1minusR(120582minus120585)R(120585) gt minus1 and let120582119896minus120585119896+120592 = 1 2 sdot sdot sdot then the following result holds

(119868(120592)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120582119896)minus(120585119896)+120592minus1

(2119896)(120585119896)sdot Γ (1 minus 120592 + (120585 minus 120582) 119896)Γ (120585119896 + 1) Γ (1 + (120585 minus 120582) 119896)

times 21198653 [[[

1 minus 1205832 + 120585 minus 120582

2119896 2 minus 1205922 + 120585 minus 120582

2119896 120585119896 + 1 12 + 120585 minus 120582

2119896 1 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]](40)

Corollary 17 Let 120592 120574 120585 120582 isin C and 119896 gt 0 be such thatR(120585) gtminus1R(120592) gt 0R(120582+120585) lt 1+max[0R(120574)] and let 120582119896minus120585119896minus120574 = 1 2 sdot sdot sdot then the following formula holds

(119870minus(120592120574)119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120582119896)minus(120585119896)minus1

(2119896)(120585119896)Γ ((120585 minus 120582) 119896 + 120574 + 1)

Γ (1 + (120585 minus 120582) 119896) Γ (120585119896 + 1)

times 21198653 [[[

120574 + 12 + 120585 minus 120582

2119896 120574 + 22 + 120585 minus 120582

2119896 120585119896 + 1 120592 + 120574 + 1

2 + 120585 minus 120582119896 120592 + 120574 + 2

2 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]]

(41)

4 Concluding Remark

MSM fractional integral operators have advantage that theygeneralize theR-LWeyl Erdelyi-Kober and Saigorsquos fractionalintegral operators therefore several authors called this ageneral operator So we conclude this paper by emphasizingthat many other interesting image formulas can be derivedas the specific cases of our leading results Theorems 2 and6 involving familiar fractional integral operators as abovesaid Further the generalized Bessel function defined in (1)

possesses the lead that a number of Bessel functions trigono-metric functions and hyperbolic functions happen to bethe particular cases of this function Some special cases ofintegrals involving generalized Bessel function have beenexplored in the literature by a number of authors ([20ndash26])with different arguments Therefore results presented in thispaper are easily converted in terms of a comparable typeof novel interesting integrals with diverse arguments aftervarious suitable parametric replacements

8 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] S R Mondal Representation Formulae and Monotonicity ofthe Generalized k-Bessel Functions 2016

[2] Diaz and E Pariguan ldquoOn hypergeometric functions andPochhammer k-symbolrdquo Divulgaciones Matematicas vol 15no 2 pp 179ndash192 2007

[3] H SatohHigher Transcendental Functions vol 2McGraw-HillNewYork Toronto London 1953

[4] P AgarwalM Chand J Choi andG Singh ldquoCertain fractionalintegrals and image formulas of generalized k-Bessel functionrdquoCommunications of the KoreanMathematical Society vol 33 no2 pp 423ndash436 2018

[5] M Chand P Agarwal and Z Hammouch ldquoCertain SequencesInvolving Product of k-Bessel Functionrdquo International Journalof Applied and Computational Mathematics vol 4 no 4 4101pages 2018

[6] K S Gehlot ldquoDifferential Equation of K -Bessels Functions andits Propertiesrdquo Nonl Analysis and Differential Equations vol 2no 2 pp 61ndash67 2014

[7] K S Gehlot ldquoRecurrence Relations of K -Bessels functionrdquoThaiJ Math

[8] K S Gehlot and S D Purohit ldquoIntegral representations of thek-Besselrsquos functionrdquo Honam Mathematical Journal vol 38 no1 pp 17ndash23 2016

[9] G Singh P AgarwalM Chand and S Jain ldquo Certain fractionalkinetic equations involving generalized rdquo Transactions of ARazmadze Mathematical Institute 2018

[10] M Saigo and N Maeda ldquoMore generalization of fractionalcalculusrdquo in TransformMethods and Special Functions pp 386ndash400 Bulgarian Acad Sci Sofia 1998

[11] O I Marichev ldquoVolterra equation of Mellin convolution typewith a Horn function in the kernelrdquo Izvestiya Akademii NaukBSSR Seriya Fiziko-Matematicheskikh Nauk vol 1 pp 128-1291974 (Russian)

[12] H M Srivastava and P W Karlsson Multiple Gaussian hyper-geometric series Ellis Horwood Series Mathematics and itsApplications Ellis Horwood Ltd Chichester Halsted Press[John Wiley amp Sons Inc] New York NY USA 1985

[13] F W J Olver DW Lozier R F Boisvert and CW Clark EdsNIST Handbook of Mathematical Functions National Instituteof Standards and Technology USA Cambridge University PressGaithersburg Md 2010

[14] A Erdersquolyi W Magnus F Oberhettinger and F G Tri-comi Higher Transcendental Functions vol 1 McGraw-HillNewYork Toronto London 1953

[15] E M Wright ldquoThe asymptotic expansion of the generalizedhypergeometric functionrdquo Journal of the London MathematicalSociety vol 1-10 no 4 pp 286ndash293 1935

[16] E M Wright ldquoThe asymptotic expansion of the generalizedhypergeometric functionrdquo Proceedings of the London Mathe-matical Society vol 46 no 2 pp 389ndash408 1940

[17] E M Wright ldquoThe asymptotic expansion of integral functionsdefined by Taylor seriesrdquo Philosophical Transactions of the RoyalSociety A Mathematical Physical amp Engineering Sciences vol238 pp 423ndash451 1940

[18] A A Kilbas M Saigo and J J Trujillo ldquoOn the generalizedWright functionrdquo Fractional Calculus and Applied Analysis vol5 no 4 pp 437ndash460 2002

[19] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash1431978

[20] A A Kilbas and N Sebastian ldquoGeneralized fractional integra-tion of Bessel function of the first kindrdquo Integral Transforms andSpecial Functions vol 19 no 11-12 pp 869ndash883 2008

[21] H Amsalu and D L Suthar ldquoGeneralized Fractional IntegralOperators Involving Mittag-Leffler Functionrdquo Abstract andApplied Analysis vol 2018 Article ID 7034124 8 pages 2018

[22] S R Mondal and K S Nisar ldquoMarichev-Saigo-Maeda Frac-tional Integration Operators Involving Generalized BesselFunctionsrdquo Mathematical Problems in Engineering vol 2014Article ID 274093 11 pages 2014

[23] K S Nisar D L Suthar S D Purohit and M AldhaifallahldquoSome unified integrals associated with the generalized Struvefunctionrdquo Proceedings of the Jangjeon Mathematical SocietyMemoirs of the Jangjeon Mathematical Society vol 20 no 2 pp261ndash267 2017

[24] S D Purohit D L Suthar and S L Kalla ldquoMarichev-Saigo-Maeda fractional integration operators of the Bessel functionsrdquoLe Matematiche vol 67 no 1 pp 21ndash32 2012

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

[26] D L Suthar and H Habenom ldquoIntegrals involving generalizedBessel-Maitland functionrdquo Journal of Science andArts no 4(37)pp 357ndash362 2016

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6 Journal of Mathematics

(120582 + 120585)119896 + 120574 minus 120583 = 0 minus1 sdot sdot sdot then the following resultholds

(119868(120592120583120574)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120585119896)+(120582119896)minus120583minus1(2119896)(120585119896)

Γ ((120582 + 120585) 119896) Γ ((120582 + 120585) 119896 + 120574 minus 120583)Γ (V119896 + 1) Γ ((120582 + 120585) 119896 minus 120583) Γ ((120582 + 120585) 119896 + 120592 + 120574)

times41198655 [[[

120582 + 1205852119896 120582 + 120585

2119896 + 12

120582 + 1205852119896 + 120574 minus 120583

2 120582 + 1205852119896 + 1 + 120574 minus 120583

2 V119896 + 1 120582 + 120585

2119896 minus 1205832

120582 + 1205852119896 minus 120583 + 1

2 120582 + 120585119896 + 120592 + 120574

2 120582 + 1205852119896 + 1 + 120592 + 120582

2 100381610038161003816100381610038161003816100381610038161003816minus

11988811990924119896 ]]

]

(34)

Corollary 12 Let 120592 120582 120585 isin C be such that R(120592) gt 0 R(120582 +120585) gt 0 R(120585) gt minus1 and 120582119896 + 120585119896 = 0 minus1 sdot sdot sdot then thefollowing result holds

(119868(120592)0+ 119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120582119896)+(120585119896)+120592minus1(2119896)(120585119896)

sdot Γ ((120582 + 120585) 119896)Γ (1 + 120585119896) Γ ((120582 + 120585) 119896 minus 120583)

times 21198653 [[[

120582 + 1205852119896 120582 + 120585

2119896 + 12 120585

119896 + 1 120582 + 1205852119896 minus 120583

2 120582 + 1205852119896 minus 120583 + 1

2 100381610038161003816100381610038161003816100381610038161003816minus

11988811990924119896 ]]

](35)

Corollary 13 Let 120592 120574 120585 120582 isin C be such thatR(120592) gt 0R(120582 +120585) gt 0R(120585) gt minus1 and let 120582119896 + 120585119896 + 120574 minus 120583 = 0 minus1 sdot sdot sdot thenthe following result holds

(119870+(120592120574)119905(120582119896)minus1119882119896120585119888 (119905)) (119909) = 119909(120585119896)+(120582119896)minus1(2119896)(120585119896)

sdot Γ ((120582 + 120585) 119896 + 120574)Γ (1 + 120585119896) Γ ((120582 + 120585) 119896 + 120592 + 120574)

times21198653 [[[

120582 + 1205852119896 + 120574

2 120582 + 1205852119896 + 120574 + 1

2120585119896 + 1 120582 + 120585

119896 + 120574 + 1205922 120582 + 120585

2119896 + 1 + 120592 + 1205742

100381610038161003816100381610038161003816100381610038161003816minus11988811990924119896 ]]

]

(36)

Theorem 14 Letting 120592 1205921015840 120583 1205831015840 120578 120582 V isin C be such that 119896 gt 0R(120574) gt 0R(]) gt minus1R(120582 minus V minus 2119899 minus 1) le 1 +R(120582 minus V minus 1) lt1+minR(minus120583)R(120592+ 1205921015840 minus120578)R(120592+ 1205831015840 minus120578) then there holdsthe following formula

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896V119888 (1119905 )) (119909) = 119909((120582minus])119896)+120578minus120592minus1205921015840minus1

(2119896)(]119896) times Γ (1 + (] minus 120582) 119896 minus 120583) Γ (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840) Γ (1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578)Γ (V119896 + 1) Γ (1 + (] minus 120582) 119896) Γ (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 + 120592 minus 120583)

times61198657 [[[[

1 + 120592 + 1205921015840 minus 1205782 + ] minus 120582

2119896 1 + ] minus 1205822119896 + 120592 + 1205921015840 minus 120578

2 1 + 120592 + 1205831015840 minus 1205782 + ] minus 120582

2119896 1 + ] minus 1205822119896 + 120592 + 1205831015840 minus 120578

2 1 minus 1205832 + ] minus 120582

2119896 1 + ] minus 1205822119896 minus 120583

2 V119896 + 1 1 + 120592 + 1205921015840 + 1205831015840 minus 120578

2 + ] minus 1205822119896 1 + ] minus 120582

2119896 + 120592 + 1205921015840 + 1205831015840 minus 1205782 12 + ] minus 120582

2119896 1 + ] minus 1205822119896 1 + 120592 minus 120583

2 + ] minus 1205822119896 1 + ] minus 120582

2119896 + 120592 minus 1205832

1003816100381610038161003816100381610038161003816minus119888

41198961199092]]]](37)

Proof Using (11) with 119911 = V119896 + 1 and (26) and applying (31)with 119911 being replaced by (]minus120582)119896minus120583+1 (]minus120582)119896+120592+1205831015840minus120578+1and (] minus 120582)119896 minus 120578 + 120592 + 1205921015840 + 1 we have

(119868(12059212059210158401205831205831015840120578)minus 119905(120582119896)minus1119882119896V119888 (1119905 )) (119909) = 119909((120582minus])119896)+120578minus120592minus1205921015840minus1

(2119896)(]119896)

times infinsum119899=0

Γ (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840) Γ (1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 minus 120583)Γ (1 + (] minus 120582) 119896) Γ (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 + 120592 minus 120583) Γ (V119896 + 1)

times (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840)2119899

(1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578)2119899

(1 + (] minus 120582) 119896 minus 120583)2119899(1 + (] minus 120582) 119896)2119899 (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578)2119899 (1 + (] minus 120582) 119896 + 120592 minus 120583)2119899 (V119896 + 1)119899

(minus119888)119899(41198961199092)119899 119899

= 119909((120582minus])119896)+120578minus120592minus1205921015840minus1(2119896)(]119896)

Γ (1 + (] minus 120582) 119896 minus 120578 + 120592 + 1205921015840) Γ (1 + (] minus 120582) 119896 + 120592 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 minus 120583)Γ (1 + (] minus 120582) 119896) Γ (1 + (] minus 120582) 119896 + 120592 + 1205921015840 + 1205831015840 minus 120578) Γ (1 + (] minus 120582) 119896 + 120592 minus 120583) Γ (V119896 + 1)

times infinsum119899=0

(12 + (] minus 120582) 2119896 + (120592 + 1205921015840 minus 120578) 2)119899(1 + (] minus 120582) 2119896 + (120592 + 1205921015840 minus 120578) 2)

119899(V119896 + 1)119899 (12 + (] minus 120582) 2119896 + (120592 + 1205921015840 + 1205831015840 minus 120578) 2)119899 (1 + (] minus 120582) 2119896 + (120592 + 1205921015840 + 1205831015840 minus 120578) 2)119899

Journal of Mathematics 7

times (12 + (] minus 120582) 2119896 + (120592 + 1205831015840 minus 120578) 2)119899(1 + (] minus 120582) 2119896 + (120592 + 1205831015840 minus 120578) 2)

119899(12 + (] minus 120582) 2119896 minus 1205832)119899 (1 + (] minus 120582) 2119896 minus 1205832)119899

(12 + (] minus 120582) 2119896)119899 (1 + (] minus 120582) 2119896)119899 (1 + (] minus 120582) 2119896 + (120592 minus 120583) 2)119899 (12 + (] minus 120582) 2119896 + (120592 minus 120583) 2)119899sdot (minus119888)119899(41198961199092)119899 119899

(38)

Thus in accordance with (9) we get the required result (37)

Corollary 15 Let 120592 120583 120574 120585 120582 isin C and 119896 gt 0 be such thatR(120592) gt 0R(120585) gt minus1 R(V) gt minus1R(120592) gt 0R(120582 minus

V) lt 1 + min[R(120583)R(120574)] and let (120583 minus 120582)119896 + 120585119896 +1 120574 minus 120582119896 + 120585119896 + 1 = 0 minus1 sdot sdot sdot then the following resultholds

(119868(120592120583120574)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909)

= 119909(120582119896)minus(120585119896)minus120583minus1(2119896)(120585119896) times Γ ((120585 minus 120582) 119896 + 120574 + 1) Γ ((120585 minus 120582) 119896 + 120583 + 1)

Γ (120585119896 + 1) Γ (1 + (120585 minus 120582) 119896) Γ ((120585 minus 120582) 119896 + 120592 + 120583 + 120578 + 1)

times41198655 [[[

120583 + 12 + 120585 minus 120582

2119896 120583 + 22 + 120585 minus 120582

2119896 120574 + 12 + 120585 minus 120582

2119896 120574 + 22 + 120585 minus 120582

2119896 V119896 + 1 12 + 120585 minus 120582

2119896 1 + 120585 minus 1205822119896 120592 + 120583 + 120574 + 1

2 + 120585 minus 120582119896 120592 + 120574 + 2

2 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]]

(39)

Corollary 16 Let 120592 120574 120585 120582 isin C and 119896 gt 0 be such that 0 ltR(120592) lt 1minusR(120582minus120585)R(120585) gt minus1 and let120582119896minus120585119896+120592 = 1 2 sdot sdot sdot then the following result holds

(119868(120592)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120582119896)minus(120585119896)+120592minus1

(2119896)(120585119896)sdot Γ (1 minus 120592 + (120585 minus 120582) 119896)Γ (120585119896 + 1) Γ (1 + (120585 minus 120582) 119896)

times 21198653 [[[

1 minus 1205832 + 120585 minus 120582

2119896 2 minus 1205922 + 120585 minus 120582

2119896 120585119896 + 1 12 + 120585 minus 120582

2119896 1 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]](40)

Corollary 17 Let 120592 120574 120585 120582 isin C and 119896 gt 0 be such thatR(120585) gtminus1R(120592) gt 0R(120582+120585) lt 1+max[0R(120574)] and let 120582119896minus120585119896minus120574 = 1 2 sdot sdot sdot then the following formula holds

(119870minus(120592120574)119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120582119896)minus(120585119896)minus1

(2119896)(120585119896)Γ ((120585 minus 120582) 119896 + 120574 + 1)

Γ (1 + (120585 minus 120582) 119896) Γ (120585119896 + 1)

times 21198653 [[[

120574 + 12 + 120585 minus 120582

2119896 120574 + 22 + 120585 minus 120582

2119896 120585119896 + 1 120592 + 120574 + 1

2 + 120585 minus 120582119896 120592 + 120574 + 2

2 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]]

(41)

4 Concluding Remark

MSM fractional integral operators have advantage that theygeneralize theR-LWeyl Erdelyi-Kober and Saigorsquos fractionalintegral operators therefore several authors called this ageneral operator So we conclude this paper by emphasizingthat many other interesting image formulas can be derivedas the specific cases of our leading results Theorems 2 and6 involving familiar fractional integral operators as abovesaid Further the generalized Bessel function defined in (1)

possesses the lead that a number of Bessel functions trigono-metric functions and hyperbolic functions happen to bethe particular cases of this function Some special cases ofintegrals involving generalized Bessel function have beenexplored in the literature by a number of authors ([20ndash26])with different arguments Therefore results presented in thispaper are easily converted in terms of a comparable typeof novel interesting integrals with diverse arguments aftervarious suitable parametric replacements

8 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] S R Mondal Representation Formulae and Monotonicity ofthe Generalized k-Bessel Functions 2016

[2] Diaz and E Pariguan ldquoOn hypergeometric functions andPochhammer k-symbolrdquo Divulgaciones Matematicas vol 15no 2 pp 179ndash192 2007

[3] H SatohHigher Transcendental Functions vol 2McGraw-HillNewYork Toronto London 1953

[4] P AgarwalM Chand J Choi andG Singh ldquoCertain fractionalintegrals and image formulas of generalized k-Bessel functionrdquoCommunications of the KoreanMathematical Society vol 33 no2 pp 423ndash436 2018

[5] M Chand P Agarwal and Z Hammouch ldquoCertain SequencesInvolving Product of k-Bessel Functionrdquo International Journalof Applied and Computational Mathematics vol 4 no 4 4101pages 2018

[6] K S Gehlot ldquoDifferential Equation of K -Bessels Functions andits Propertiesrdquo Nonl Analysis and Differential Equations vol 2no 2 pp 61ndash67 2014

[7] K S Gehlot ldquoRecurrence Relations of K -Bessels functionrdquoThaiJ Math

[8] K S Gehlot and S D Purohit ldquoIntegral representations of thek-Besselrsquos functionrdquo Honam Mathematical Journal vol 38 no1 pp 17ndash23 2016

[9] G Singh P AgarwalM Chand and S Jain ldquo Certain fractionalkinetic equations involving generalized rdquo Transactions of ARazmadze Mathematical Institute 2018

[10] M Saigo and N Maeda ldquoMore generalization of fractionalcalculusrdquo in TransformMethods and Special Functions pp 386ndash400 Bulgarian Acad Sci Sofia 1998

[11] O I Marichev ldquoVolterra equation of Mellin convolution typewith a Horn function in the kernelrdquo Izvestiya Akademii NaukBSSR Seriya Fiziko-Matematicheskikh Nauk vol 1 pp 128-1291974 (Russian)

[12] H M Srivastava and P W Karlsson Multiple Gaussian hyper-geometric series Ellis Horwood Series Mathematics and itsApplications Ellis Horwood Ltd Chichester Halsted Press[John Wiley amp Sons Inc] New York NY USA 1985

[13] F W J Olver DW Lozier R F Boisvert and CW Clark EdsNIST Handbook of Mathematical Functions National Instituteof Standards and Technology USA Cambridge University PressGaithersburg Md 2010

[14] A Erdersquolyi W Magnus F Oberhettinger and F G Tri-comi Higher Transcendental Functions vol 1 McGraw-HillNewYork Toronto London 1953

[15] E M Wright ldquoThe asymptotic expansion of the generalizedhypergeometric functionrdquo Journal of the London MathematicalSociety vol 1-10 no 4 pp 286ndash293 1935

[16] E M Wright ldquoThe asymptotic expansion of the generalizedhypergeometric functionrdquo Proceedings of the London Mathe-matical Society vol 46 no 2 pp 389ndash408 1940

[17] E M Wright ldquoThe asymptotic expansion of integral functionsdefined by Taylor seriesrdquo Philosophical Transactions of the RoyalSociety A Mathematical Physical amp Engineering Sciences vol238 pp 423ndash451 1940

[18] A A Kilbas M Saigo and J J Trujillo ldquoOn the generalizedWright functionrdquo Fractional Calculus and Applied Analysis vol5 no 4 pp 437ndash460 2002

[19] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash1431978

[20] A A Kilbas and N Sebastian ldquoGeneralized fractional integra-tion of Bessel function of the first kindrdquo Integral Transforms andSpecial Functions vol 19 no 11-12 pp 869ndash883 2008

[21] H Amsalu and D L Suthar ldquoGeneralized Fractional IntegralOperators Involving Mittag-Leffler Functionrdquo Abstract andApplied Analysis vol 2018 Article ID 7034124 8 pages 2018

[22] S R Mondal and K S Nisar ldquoMarichev-Saigo-Maeda Frac-tional Integration Operators Involving Generalized BesselFunctionsrdquo Mathematical Problems in Engineering vol 2014Article ID 274093 11 pages 2014

[23] K S Nisar D L Suthar S D Purohit and M AldhaifallahldquoSome unified integrals associated with the generalized Struvefunctionrdquo Proceedings of the Jangjeon Mathematical SocietyMemoirs of the Jangjeon Mathematical Society vol 20 no 2 pp261ndash267 2017

[24] S D Purohit D L Suthar and S L Kalla ldquoMarichev-Saigo-Maeda fractional integration operators of the Bessel functionsrdquoLe Matematiche vol 67 no 1 pp 21ndash32 2012

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

[26] D L Suthar and H Habenom ldquoIntegrals involving generalizedBessel-Maitland functionrdquo Journal of Science andArts no 4(37)pp 357ndash362 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Journal of Mathematics 7

times (12 + (] minus 120582) 2119896 + (120592 + 1205831015840 minus 120578) 2)119899(1 + (] minus 120582) 2119896 + (120592 + 1205831015840 minus 120578) 2)

119899(12 + (] minus 120582) 2119896 minus 1205832)119899 (1 + (] minus 120582) 2119896 minus 1205832)119899

(12 + (] minus 120582) 2119896)119899 (1 + (] minus 120582) 2119896)119899 (1 + (] minus 120582) 2119896 + (120592 minus 120583) 2)119899 (12 + (] minus 120582) 2119896 + (120592 minus 120583) 2)119899sdot (minus119888)119899(41198961199092)119899 119899

(38)

Thus in accordance with (9) we get the required result (37)

Corollary 15 Let 120592 120583 120574 120585 120582 isin C and 119896 gt 0 be such thatR(120592) gt 0R(120585) gt minus1 R(V) gt minus1R(120592) gt 0R(120582 minus

V) lt 1 + min[R(120583)R(120574)] and let (120583 minus 120582)119896 + 120585119896 +1 120574 minus 120582119896 + 120585119896 + 1 = 0 minus1 sdot sdot sdot then the following resultholds

(119868(120592120583120574)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909)

= 119909(120582119896)minus(120585119896)minus120583minus1(2119896)(120585119896) times Γ ((120585 minus 120582) 119896 + 120574 + 1) Γ ((120585 minus 120582) 119896 + 120583 + 1)

Γ (120585119896 + 1) Γ (1 + (120585 minus 120582) 119896) Γ ((120585 minus 120582) 119896 + 120592 + 120583 + 120578 + 1)

times41198655 [[[

120583 + 12 + 120585 minus 120582

2119896 120583 + 22 + 120585 minus 120582

2119896 120574 + 12 + 120585 minus 120582

2119896 120574 + 22 + 120585 minus 120582

2119896 V119896 + 1 12 + 120585 minus 120582

2119896 1 + 120585 minus 1205822119896 120592 + 120583 + 120574 + 1

2 + 120585 minus 120582119896 120592 + 120574 + 2

2 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]]

(39)

Corollary 16 Let 120592 120574 120585 120582 isin C and 119896 gt 0 be such that 0 ltR(120592) lt 1minusR(120582minus120585)R(120585) gt minus1 and let120582119896minus120585119896+120592 = 1 2 sdot sdot sdot then the following result holds

(119868(120592)minus 119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120582119896)minus(120585119896)+120592minus1

(2119896)(120585119896)sdot Γ (1 minus 120592 + (120585 minus 120582) 119896)Γ (120585119896 + 1) Γ (1 + (120585 minus 120582) 119896)

times 21198653 [[[

1 minus 1205832 + 120585 minus 120582

2119896 2 minus 1205922 + 120585 minus 120582

2119896 120585119896 + 1 12 + 120585 minus 120582

2119896 1 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]](40)

Corollary 17 Let 120592 120574 120585 120582 isin C and 119896 gt 0 be such thatR(120585) gtminus1R(120592) gt 0R(120582+120585) lt 1+max[0R(120574)] and let 120582119896minus120585119896minus120574 = 1 2 sdot sdot sdot then the following formula holds

(119870minus(120592120574)119905(120582119896)minus1119882119896120585119888 (1119905 )) (119909) = 119909(120582119896)minus(120585119896)minus1

(2119896)(120585119896)Γ ((120585 minus 120582) 119896 + 120574 + 1)

Γ (1 + (120585 minus 120582) 119896) Γ (120585119896 + 1)

times 21198653 [[[

120574 + 12 + 120585 minus 120582

2119896 120574 + 22 + 120585 minus 120582

2119896 120585119896 + 1 120592 + 120574 + 1

2 + 120585 minus 120582119896 120592 + 120574 + 2

2 + 120585 minus 1205822119896

1003816100381610038161003816100381610038161003816minus119888

41198961199092 ]]]

(41)

4 Concluding Remark

MSM fractional integral operators have advantage that theygeneralize theR-LWeyl Erdelyi-Kober and Saigorsquos fractionalintegral operators therefore several authors called this ageneral operator So we conclude this paper by emphasizingthat many other interesting image formulas can be derivedas the specific cases of our leading results Theorems 2 and6 involving familiar fractional integral operators as abovesaid Further the generalized Bessel function defined in (1)

possesses the lead that a number of Bessel functions trigono-metric functions and hyperbolic functions happen to bethe particular cases of this function Some special cases ofintegrals involving generalized Bessel function have beenexplored in the literature by a number of authors ([20ndash26])with different arguments Therefore results presented in thispaper are easily converted in terms of a comparable typeof novel interesting integrals with diverse arguments aftervarious suitable parametric replacements

8 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] S R Mondal Representation Formulae and Monotonicity ofthe Generalized k-Bessel Functions 2016

[2] Diaz and E Pariguan ldquoOn hypergeometric functions andPochhammer k-symbolrdquo Divulgaciones Matematicas vol 15no 2 pp 179ndash192 2007

[3] H SatohHigher Transcendental Functions vol 2McGraw-HillNewYork Toronto London 1953

[4] P AgarwalM Chand J Choi andG Singh ldquoCertain fractionalintegrals and image formulas of generalized k-Bessel functionrdquoCommunications of the KoreanMathematical Society vol 33 no2 pp 423ndash436 2018

[5] M Chand P Agarwal and Z Hammouch ldquoCertain SequencesInvolving Product of k-Bessel Functionrdquo International Journalof Applied and Computational Mathematics vol 4 no 4 4101pages 2018

[6] K S Gehlot ldquoDifferential Equation of K -Bessels Functions andits Propertiesrdquo Nonl Analysis and Differential Equations vol 2no 2 pp 61ndash67 2014

[7] K S Gehlot ldquoRecurrence Relations of K -Bessels functionrdquoThaiJ Math

[8] K S Gehlot and S D Purohit ldquoIntegral representations of thek-Besselrsquos functionrdquo Honam Mathematical Journal vol 38 no1 pp 17ndash23 2016

[9] G Singh P AgarwalM Chand and S Jain ldquo Certain fractionalkinetic equations involving generalized rdquo Transactions of ARazmadze Mathematical Institute 2018

[10] M Saigo and N Maeda ldquoMore generalization of fractionalcalculusrdquo in TransformMethods and Special Functions pp 386ndash400 Bulgarian Acad Sci Sofia 1998

[11] O I Marichev ldquoVolterra equation of Mellin convolution typewith a Horn function in the kernelrdquo Izvestiya Akademii NaukBSSR Seriya Fiziko-Matematicheskikh Nauk vol 1 pp 128-1291974 (Russian)

[12] H M Srivastava and P W Karlsson Multiple Gaussian hyper-geometric series Ellis Horwood Series Mathematics and itsApplications Ellis Horwood Ltd Chichester Halsted Press[John Wiley amp Sons Inc] New York NY USA 1985

[13] F W J Olver DW Lozier R F Boisvert and CW Clark EdsNIST Handbook of Mathematical Functions National Instituteof Standards and Technology USA Cambridge University PressGaithersburg Md 2010

[14] A Erdersquolyi W Magnus F Oberhettinger and F G Tri-comi Higher Transcendental Functions vol 1 McGraw-HillNewYork Toronto London 1953

[15] E M Wright ldquoThe asymptotic expansion of the generalizedhypergeometric functionrdquo Journal of the London MathematicalSociety vol 1-10 no 4 pp 286ndash293 1935

[16] E M Wright ldquoThe asymptotic expansion of the generalizedhypergeometric functionrdquo Proceedings of the London Mathe-matical Society vol 46 no 2 pp 389ndash408 1940

[17] E M Wright ldquoThe asymptotic expansion of integral functionsdefined by Taylor seriesrdquo Philosophical Transactions of the RoyalSociety A Mathematical Physical amp Engineering Sciences vol238 pp 423ndash451 1940

[18] A A Kilbas M Saigo and J J Trujillo ldquoOn the generalizedWright functionrdquo Fractional Calculus and Applied Analysis vol5 no 4 pp 437ndash460 2002

[19] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash1431978

[20] A A Kilbas and N Sebastian ldquoGeneralized fractional integra-tion of Bessel function of the first kindrdquo Integral Transforms andSpecial Functions vol 19 no 11-12 pp 869ndash883 2008

[21] H Amsalu and D L Suthar ldquoGeneralized Fractional IntegralOperators Involving Mittag-Leffler Functionrdquo Abstract andApplied Analysis vol 2018 Article ID 7034124 8 pages 2018

[22] S R Mondal and K S Nisar ldquoMarichev-Saigo-Maeda Frac-tional Integration Operators Involving Generalized BesselFunctionsrdquo Mathematical Problems in Engineering vol 2014Article ID 274093 11 pages 2014

[23] K S Nisar D L Suthar S D Purohit and M AldhaifallahldquoSome unified integrals associated with the generalized Struvefunctionrdquo Proceedings of the Jangjeon Mathematical SocietyMemoirs of the Jangjeon Mathematical Society vol 20 no 2 pp261ndash267 2017

[24] S D Purohit D L Suthar and S L Kalla ldquoMarichev-Saigo-Maeda fractional integration operators of the Bessel functionsrdquoLe Matematiche vol 67 no 1 pp 21ndash32 2012

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

[26] D L Suthar and H Habenom ldquoIntegrals involving generalizedBessel-Maitland functionrdquo Journal of Science andArts no 4(37)pp 357ndash362 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Journal of Mathematics

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] S R Mondal Representation Formulae and Monotonicity ofthe Generalized k-Bessel Functions 2016

[2] Diaz and E Pariguan ldquoOn hypergeometric functions andPochhammer k-symbolrdquo Divulgaciones Matematicas vol 15no 2 pp 179ndash192 2007

[3] H SatohHigher Transcendental Functions vol 2McGraw-HillNewYork Toronto London 1953

[4] P AgarwalM Chand J Choi andG Singh ldquoCertain fractionalintegrals and image formulas of generalized k-Bessel functionrdquoCommunications of the KoreanMathematical Society vol 33 no2 pp 423ndash436 2018

[5] M Chand P Agarwal and Z Hammouch ldquoCertain SequencesInvolving Product of k-Bessel Functionrdquo International Journalof Applied and Computational Mathematics vol 4 no 4 4101pages 2018

[6] K S Gehlot ldquoDifferential Equation of K -Bessels Functions andits Propertiesrdquo Nonl Analysis and Differential Equations vol 2no 2 pp 61ndash67 2014

[7] K S Gehlot ldquoRecurrence Relations of K -Bessels functionrdquoThaiJ Math

[8] K S Gehlot and S D Purohit ldquoIntegral representations of thek-Besselrsquos functionrdquo Honam Mathematical Journal vol 38 no1 pp 17ndash23 2016

[9] G Singh P AgarwalM Chand and S Jain ldquo Certain fractionalkinetic equations involving generalized rdquo Transactions of ARazmadze Mathematical Institute 2018

[10] M Saigo and N Maeda ldquoMore generalization of fractionalcalculusrdquo in TransformMethods and Special Functions pp 386ndash400 Bulgarian Acad Sci Sofia 1998

[11] O I Marichev ldquoVolterra equation of Mellin convolution typewith a Horn function in the kernelrdquo Izvestiya Akademii NaukBSSR Seriya Fiziko-Matematicheskikh Nauk vol 1 pp 128-1291974 (Russian)

[12] H M Srivastava and P W Karlsson Multiple Gaussian hyper-geometric series Ellis Horwood Series Mathematics and itsApplications Ellis Horwood Ltd Chichester Halsted Press[John Wiley amp Sons Inc] New York NY USA 1985

[13] F W J Olver DW Lozier R F Boisvert and CW Clark EdsNIST Handbook of Mathematical Functions National Instituteof Standards and Technology USA Cambridge University PressGaithersburg Md 2010

[14] A Erdersquolyi W Magnus F Oberhettinger and F G Tri-comi Higher Transcendental Functions vol 1 McGraw-HillNewYork Toronto London 1953

[15] E M Wright ldquoThe asymptotic expansion of the generalizedhypergeometric functionrdquo Journal of the London MathematicalSociety vol 1-10 no 4 pp 286ndash293 1935

[16] E M Wright ldquoThe asymptotic expansion of the generalizedhypergeometric functionrdquo Proceedings of the London Mathe-matical Society vol 46 no 2 pp 389ndash408 1940

[17] E M Wright ldquoThe asymptotic expansion of integral functionsdefined by Taylor seriesrdquo Philosophical Transactions of the RoyalSociety A Mathematical Physical amp Engineering Sciences vol238 pp 423ndash451 1940

[18] A A Kilbas M Saigo and J J Trujillo ldquoOn the generalizedWright functionrdquo Fractional Calculus and Applied Analysis vol5 no 4 pp 437ndash460 2002

[19] M Saigo ldquoA remark on integral operators involving the Gausshypergeometric functionsrdquo Mathematical Reports of College ofGeneral Education Kyushu University vol 11 no 2 pp 135ndash1431978

[20] A A Kilbas and N Sebastian ldquoGeneralized fractional integra-tion of Bessel function of the first kindrdquo Integral Transforms andSpecial Functions vol 19 no 11-12 pp 869ndash883 2008

[21] H Amsalu and D L Suthar ldquoGeneralized Fractional IntegralOperators Involving Mittag-Leffler Functionrdquo Abstract andApplied Analysis vol 2018 Article ID 7034124 8 pages 2018

[22] S R Mondal and K S Nisar ldquoMarichev-Saigo-Maeda Frac-tional Integration Operators Involving Generalized BesselFunctionsrdquo Mathematical Problems in Engineering vol 2014Article ID 274093 11 pages 2014

[23] K S Nisar D L Suthar S D Purohit and M AldhaifallahldquoSome unified integrals associated with the generalized Struvefunctionrdquo Proceedings of the Jangjeon Mathematical SocietyMemoirs of the Jangjeon Mathematical Society vol 20 no 2 pp261ndash267 2017

[24] S D Purohit D L Suthar and S L Kalla ldquoMarichev-Saigo-Maeda fractional integration operators of the Bessel functionsrdquoLe Matematiche vol 67 no 1 pp 21ndash32 2012

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

[26] D L Suthar and H Habenom ldquoIntegrals involving generalizedBessel-Maitland functionrdquo Journal of Science andArts no 4(37)pp 357ndash362 2016

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

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