Research Article Marichev-Saigo-Maeda Fractional Integration Operators Involving Generalized Bessel Functions

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1 Mahemaical Problems in Engineering Volume 014 Aricle ID pages hp://dx.doi.org/ /014/74093 Research Aricle Marichev-Saigo-Maeda Fracional Inegraion Operaors Involving Generalized Bessel Funcions Saiful R. Mondal 1 andk.s.nisar 1 Deparmen of Mahemaics & Saisics College of Science King Faisal Universiy P.O. Box 400 Hofuf Al-Ahsa 3198 Saudi Arabia Deparmen of Mahemaics College of Ars and Science Salman bin Abdulaziz Universiy P.O. Box 54 Wadi Al-Dawaser Saudi Arabia Correspondence should be addressed o Saiful R. Mondal saiful786@gmail.com Received 1 February 014 Acceped 4 March 014 Published 8 April 014 Academic Edior: Sananu Saha Ray Copyrigh 014 S. R. Mondal and K. S. Nisar. This is an open access aricle disribued under he Creaive Commons Aribuion License which permis unresriced use disribuion and reproducion in any medium provided he original work is properly cied. Two inegral operaors involving Appell s funcions or Horn s funcion in he kernel are considered. Composiion of such funcions wih generalized Bessel funcions of he firs kind is expressed in erms of generalized Wrigh funcion and generalized hypergeomeric series. Many special cases including cosine and sine funcion are also discussed. 1. Inroducion Le α α β β γ C and x > 0 hen he generalized fracional inegral operaors involving Appell s funcions or Horn s funcion are defined as follows: f) (x) = x α x Γ(γ) (x ) γ 1 α F 3 0 (αα ββ γ1 x 1 x )f() d f) (x) = x α Γ(γ) ( x) γ 1 α F 3 x (αα ββ γ1 x 1 x )f() d wih Re(γ) > 0. The generalized fracional inegral operaors of he ypes (1)and()havebeeninroducedbyMarichev1 and laer exended and sudied by Saigo and Maeda. These operaors ogeher are known as he Marichev-Saigo-Maeda operaor. (1) () The fracional inegral operaor has many ineresing applicaions in various subfields in applicable mahemaical analysis for example 3 i has applicaions relaed o a cerain class of complex analyic funcions. The resuls given in 4 6 can be referred o for some basic resuls on fracional calculus. Thepurposeofhisworkisoinvesigaecomposiions of inegral ransforms (1) and() wih he generalized Bessel funcion of he firs kind W pbc defined for complex z C and b c p C by ( 1) k c k W pbc (z) = Γ (κ+k) k! (z)k+p (3) where κ:=p+(b+1)/. More deails relaed o he funcion W pbc and is paricular cases can be found in 7 8 and references herein. I is worh menioning ha W p11 =J p is Bessel funcion of order p and W p1 1 = I p is modified Bessel funcion of order p.alsow p1 =j p / π is spherical Bessel funcion of order p and W p 1 =i p / π is modified spherical Bessel funcion of order p. Thus he sudy of he inegral ransform of W pbc will give far reaching resuls han he resul in The presen paper is organized as follows. In Secions and 3 composiion of inegral ransforms (1) and() wih

2 Mahemaical Problems in Engineering generalized Bessel funcion (3) is given in erms of generalized Wrigh funcions and generalized hypergeomeric funcions respecively. Special cases like p = b/ (p =1 b/)ofw pbc give he composiion of (1)and()wihcosine and hyperbolic cosine (sine and hyperbolic sine) funcions which are discussed in Secion 4. Some concluding remarks and comparison wih earlier known work are menioned in Secion 5. The following wo resuls given by Saigo e al. 11 are needed in sequel. Lemma 1. Le α α ββ γρ C be such ha Re(γ) > 0 and Re (ρ) > max {0 Re (α α β γ)re (α β )}. (4) Then here exiss he relaion ρ 1 ) (x) where =Γ ρρ +γ α α βρ+β α ρ+β ρ+γ α α ρ+γ α β xρ α α +γ 1 (5) a b c Γ (a) Γ (b) Γ (c) Γ = d e f Γ (d) Γ (e) Γ(f). (6) Lemma. Le α α ββ γρ C be such ha Re(γ) > 0 and Re (ρ) <1+min {Re ( β) Re (α + α γ)re (α + β γ)}. (7) Then here exiss he relaion ρ 1 ) (x) =Γ 1 ρ γ+α+α 1 ρ+α+β 1 ρ β 1 ρ1 ρ +α+α +β+β γ1 ρ+α β x ρ α α +γ 1.. Represenaions in erms of Generalized Wrigh Funcions In his secion composiion of inegral ransforms (1)and() wih generalized Bessel funcion (3) isgiveninermsofhe generalized Wrigh hypergeomeric funcion p ψ q (z) which is defined by he series pψ q (z) = p ψ q (a iα i ) 1p p i=1 z= Γ(a i +α i k) z k (b j β j ) 1q q j=1 Γ(b j +β j k) k!. (9) Here a i b j C andα i β j R (i = 1...p j = 1...q). Asympoic behavior of his funcion for large values of argumen of z Cwas sudied in 1 and under he condiion q j=1 β j p i=1 (8) α i > 1 (10) in Properies of his generalized Wrigh funcion were invesigaed in In paricular i was proved 15ha pψ q (z) z C is an enire funcion under he condiion (10). Ineresing resuls relaed o generalized Wrigh funcions are also given in 17. Theorem 3. Le α α ββ γρpbc C such ha κ =1...SupposehaRe(γ) > 0 and Re(ρ+ p) > max{0 Re(α + α +β γ)re(α β )}.Then ρ 1 W pbc ()) (x) = xρ+p α α +γ 1 p 3 ψ 4 (ρ+p)(ρ+p+γ α α β)(ρ+p+β α ) (ρ+ p + β )(ρ+p+γ α α ) (ρ+p+γ α β)(κ 1) cx 4. (11) Proof. An applicaion of inegral ransform (1) o he generalized Bessel funcion (3)leadsoheformula Now changing he order of inegraion and summaion in righ-hand side of (1) yields ρ 1 W pbc ()) (x) = ( c) k (1/) k+p ρ+p+k 1 ) (x). Γ (κ+k) k! (1) ρ 1 W pbc ()) (x) = ( c) k (1/) k+p Γ (κ+k) k! ρ+p+k 1 ) (x). (13)

3 Mahemaical Problems in Engineering 3 Noe ha for all k = Re (ρ+p+k) Re (ρ+ p) > max {0 Re (α α β γ)re (α β )}. (14) Replacing ρ by ρ+p+kin Lemma 1 and using (5) we obain (I αα ββ γ ρ 1 W pbc ()) (x) = xρ+p α α +γ 1 p ρ+p+k ρ+p+γ α α Γ β+k ρ+p+β α +k ρ+p+β +k ρ+p+γ α α +k ρ+p+γ α β+kκ+k 1 k k! ( cx 4 ). (15) Inerpreing he righ-hand side of (15) he equaliy (11) canbeobainedfrom(6) and hen by using he definiion of generalized Wrigh funcion. Theorem 4. Le α α ββ γρpbc C such ha κ =1...SupposehaRe(γ) > 0 and Re(ρ p) < 1+min{Re( β) Re(α + α γ)re(α + β γ)}.then ρ 1 W pbc ( 1 )) (x) = xρ p α α +γ 1 p 3 ψ 4 (1 ρ+p γ+α+α )(1 ρ+p+α β γ)(1 ρ+p β) (1 ρ+p)(1 ρ+p γ+α+α +β )(1 ρ+p+α β)(κ k) c 4x. (16) Proof. Using () and(3) and hen changing he order of inegraion and summaion which is jusified under he condiionswihtheorem 4yield Noe ha for all 1... Re (ρ p k) Re (ρ p) ρ 1 W pbc ( 1 )) (x) = ( c) k (1/) k+p Γ (κ+k) k! ρ p k 1 ) (x). (17) <1+min {Re ( β) Re (α + α γ)re (α + β γ)}. (18) Hence replacing ρ by ρ p kin Lemma and using (8) we obain ρ 1 W pbc ( 1 )) (x) = xρ p α α +γ 1 p Γ 1 ρ+p γ+α+α +k1 ρ+p+α+β γ+k1 ρ+p β+k 1 ρ+p+k1 ρ+p γ+α+α +β +k1 ρ+p+α β+k 1 k! ( c 4x )k. (19)

4 4 Mahemaical Problems in Engineering Now (6) (9) and (19) ogeher imply ha ρ 1 W pbc ( 1 )) (x) = xρ p α α +γ 1 p 3 ψ 4 (1 ρ+p γ+α+α )(1 ρ+p+α β γ)(1 ρ+p β) (1 ρ+p)(1 ρ+p γ+α+α +β )(1 ρ+p+α β)(pk) c 4x (0) and his complees he proof. 3. Represenaion in erms of Generalized Hypergeomeric Series The generalized hypergeomeric funcion p F q (a 1... a p c 1... c q z)is given by he represenaion pf q (a 1...a p c 1...c q z) = (a 1 ) k (a p ) k z k (1) (c 1 ) k (c q ) k! (z C) k where none of he denominaor parameers is zero or a negaive ineger. Here p or q are allowed o be zero. The series (1) is convergen for all finie z if p q while for p=q+1 i is convergen for z < 1 and divergen for z > 1. Resuls obained in his secion demonsrae he image formula for he generalized Bessel funcions W pbc under he operaors (1) and() in erms of generalized hypergeomeric funcions. The well-known Legendre duplicaion formulas 18givenby Γ (z) = z 1 π Γ (z) Γ(z+ 1 ) (z) k = k ( z ) ( z+1 k ) (k N 0 ) k are required for his purpose. () Theorem 5. Le α α ββ γρpbc C such ha κ = 1... Suppose ha Re(γ) > 0 and Re(ρ+ p) > max{0 Re(α α β γ)re(α β )}. Thenhefollowing formula holds: ρ 1 W pbc ()) (x) = xρ+p 1 Γ(ρ+p)Γ(ρ+p+γ α α β) p Γ(ρ+p+β )Γ(ρ+p+γ α α )Γ(κ) ρ+p ρ+p+1 ρ+p+γ α α β ρ+p+β +1 κ ρ+p+β Γ(ρ+p+β α ) Γ(ρ+p+γ α β) ρ+p α α +β+1 ρ+p+γ α α ρ+p+γ α β ρ+p+γ α β+1 ρ+p+β α ρ+p+γ α α +1 ρ+p+β α +1 cx 4. (3) Proof. I is known ha Γ(z + k) = Γ(z)(z) k.thus ρ+p+k ρ+p+γ α α β+k ρ+p+β α +k Γ ρ+p+β +k ρ+p+γ α α +k ρ+p+γ α β+k κ+k Γ(ρ+p)Γ(ρ+p+γ α α β)γ(ρ+p+β α ) = Γ(ρ+p+β )Γ(ρ+p+γ α α )Γ(ρ+p+γ α β)γ(κ) (4) (ρ+ p) k (ρ+p+γ α α β) k (ρ+ p + β α ) k (ρ+p+β ) k (ρ+p+γ α α ) k (ρ+p+γ α β) k (κ) k.

5 Mahemaical Problems in Engineering 5 Now apply (3.3) on he righ-hand side of he above equaion and hen he resul follows from(15). This complees he proof. Theorem 6. Le α α ββ γρpbc C such ha κ =1...SupposehaRe(γ) > 0 and Re(ρ p) < 1+min{Re( β) Re(α + α γ)re(α + β γ)}.then By adoping a similar mehod he nex resul can be obained from (19) we omi he deails. ρ 1 W pbc ( 1 )) (x) Γ(α+α +p γ ρ+1)γ(α+β γ+p ρ+1)γ( β+p ρ+1) p Γ(p ρ+1)γ(κ) Γ(α+α +β +p γ ρ+1)γ(α β+p ρ+1) = xρ p α α +γ 1 α+α +p γ ρ+1 κ p ρ+1 α+α +p+γ ρ+ β+p ρ+1 α+β +p γ ρ+1 β+p ρ+ p ρ+ α+α +β +p γ ρ+1 α β+p ρ+1 α β+p ρ+ α+β +p γ ρ+ α+α +β +p γ ρ+ c 4x. (5) 4. Fracional Inegraion of Trigonomeric Funcions 4.1. Cosine and Hyperbolic Cosine Funcions. For all b Cif p = b/ hen he generalized Bessel funcion W pbc (z) has he form Hence he following resuls are a consequence of Theorems 3 and 4respecively. Corollary 7. Le α α ββ γρc C such ha Re(γ) > 0 and W b/bc (z) =( z ) b/ cos cz π W b/b c (z) =( z ) b/ cosh cz π. (6) Re (ρ) > max {0 Re (α α β γ)re (α β )}. (7) Then ρ 1 cos (c)) (x) (ρ)(ρ +γ α α β)(ρ+β α ) =π 1/ x ρ α α +γ 1 3 ψ 4 (ρ+ β )(ρ+γ α α )(ρ+γ α β)( 1 x 1) c 4 ρ 1 cosh (c)) (x) (8) (ρ)(ρ +γ α α β)(ρ+β α ) =π 1/ x ρ α α +γ 1 3 ψ 4 (ρ+ β )(ρ+γ α α )(ρ+γ α β)( 1 1) c x 4.

6 6 Mahemaical Problems in Engineering Proof. On seing p = b/ and replacing c by c ino (11) and using (37) we have ρ 1 ( b/ ) cos (c) π ) (x) = xρ (b/) α α +γ 1 (ρ b b/ 3 ψ 4 )(ρ b +γ α α β)(ρ b +β α ) (ρ b +β )(ρ b +γ α α )(ρ b +γ α β)( 1 cx 1) 4. (9) This implies ha ρ (b/) 1 cos c) (x) =π 1/ x ρ (b/) α α +γ 1 (ρ b 3 ψ 4 )(ρ b +γ α α β)(ρ b +β α ) (ρ b +β )(ρ b +γ α α )(ρ b +γ α β)( 1 cx 1) 4. (30) The ideniy (39)follows from(30)by replacing ρ by ρ+(b/). Similarly he ideniy (4.7) can be obained from (11) by seing p = b/ and replacing c by c. Corollary 8. Le α α ββ γρ C be such ha Re(γ) > 0 and Re (ρ) < min {Re ( β) Re (α + α γ)re (α + β γ)}. (31) Then ρ 1 cos ( c )) (x) (ρ γ+α+α )( ρ+α+β γ)( ρ β) =π 1/ x ρ α α +γ 3 ψ 4 ( ρ)( ρ γ+α+α +β )( ρ+α β)( 1 c 1) 4x ρ cosh ( c )) (x) (3) (ρ γ+α+α )( ρ+α+β γ)( ρ β) =π 1/ x ρ α α +γ 3 ψ 4 ( ρ)( ρ γ+α+α +β )( ρ+α β)( 1 1) c 4x. The nex saemens show ha he image formulas for cosine and hyperbolic cosine under Saigo-Maeda fracional inegral operaors can also be represened in erms of he generalized hypergeomeric series. This resul follows from Theorems 5 and 6 wih aking p = b/ and replacing c by c or c respecively. Corollary 9. Le α α ββ γρc C.SupposehaRe(γ) > 0 and Re (ρ) > max {0 Re (α α β γ)re (α β )}. (33)

7 Mahemaical Problems in Engineering 7 Then he following formula holds: ρ 1 cos (c)) (x) = Γ(ρ)Γ(ρ +γ α α β)γ(ρ+β α ) Γ(ρ+β )Γ(ρ+γ α α )Γ(ρ+γ α β) xρ α α +γ 1 ρ ρ+1 ρ+γ α α β 1 ρ+β ρ+β +1 ρ+γ α α ρ 1 cosh (c)) (x) ρ+γ α α +β+1 ρ+γ α α +1 = Γ(ρ)Γ(ρ +γ α α β)γ(ρ+β α ) Γ(ρ+β )Γ(ρ+γ α α )Γ(ρ+γ α β) xρ α α +γ 1 ρ+β α ρ+β α +1 ρ+γ α β ρ+γ α β+1 c x 4 (34) ρ ρ+1 ρ+γ α α β 1 ρ+β ρ+β +1 ρ+γ α α ρ+γ α α +β+1 ρ+γ α α +1 ρ+β α ρ+β α +1 ρ+γ α β ρ+γ α β+1 c x 4. Corollary 10. Le α α ββ γρc C. Suppose ha Re(γ) > 0 and Re (ρ) <1+min {Re ( β) Re (α + α γ)re (α + β γ)}. (35) Then ρ cos ( c )) (x) = Γ(α+α γ ρ)γ(α+β γ ρ)γ( β ρ) x ρ α α +γ Γ( ρ)γ(α+α +β γ ρ)γ(α β ρ) α+α ρ 1 ρ ρ+1 α+α γ ρ+1 α+α +β γ ρ α+β γ ρ α+β γ ρ+1 α+α +β γ ρ+1 α β ρ β ρ β ρ+ 1 α β ρ+1 c 4x (36) ρ cosh ( c )) (x) = Γ(α+α γ ρ)γ(α+β γ ρ) Γ( ρ)γ(α+α +β γ ρ) Γ( β ρ) Γ(α β ρ) xρ α α +γ α+α ρ 1 ρ ρ+1 α+α γ ρ+1 α+α +β γ ρ α+β γ ρ α+β γ ρ+1 α+α +β γ ρ+1 β ρ β ρ+ 1 α β ρ α β ρ+1 c 4x.

8 8 Mahemaical Problems in Engineering 4.. Sine and Hyperbolic Sine Funcions. For all b Cifp= 1 b/ hen he generalized Bessel funcion W pbc (z) has he form Corollary 11. Le α α ββ γρc C such ha Re(γ) > 0 and W 1 (b/)bc (z) =( z ) b/ sin (cz) π W 1 (b/)b c (z) =( b/ z ) sinh (cz). π (37) Re (ρ) > max {0 Re (α α β γ)re (α β )}. (38) Thus he composiion of Saigo-Maeda fracional inegral operaors wih sine and hyperbolic sine funcions can be obained from Theorems 3 and 4respecively. Then ρ 1 sin (c)) (x) (ρ)(ρ +γ α α β)(ρ+β α ) =π 1/ x ρ α α +γ 1 3 ψ 4 (ρ+ β )(ρ+γ α α )(ρ+γ α β)( 3 x 1) c 4 ρ 1 sinh (c)) (x) (39) (ρ)(ρ +γ α α β)(ρ+β α ) =π 1/ x ρ α α +γ 1 3 ψ 4 (ρ+ β )(ρ+γ α α )(ρ+γ α β)( 3 1) c x 4. The nex resul follows from Theorem 4 by seing p = b/ and replacing c by c or c respecively. Corollary 1. Le α α ββ γρ C be such ha Re(γ) > 0 and Re (ρ) < min {Re ( β) Re (α + α γ)re (α + β γ)}. (40) Then ρ 1 sin ( c )) (x) (ρ γ+α+α )( ρ+α+β γ)( ρ β) =π 1/ x ρ α α +γ 3 ψ 4 ( ρ)( ρ γ+α+α +β )( ρ+α β)( 3 c 1) 4x ρ sinh ( c )) (x) (41) (ρ γ+α+α )( ρ+α+β γ)( ρ β) =π 1/ x ρ α α +γ 3 ψ 4 ( ρ)( ρ γ+α+α +β )( ρ+α β)( 3 1) c 4x. The following resul can be obained from Theorems 5 and 6 wih aking p = b/ and replacing c by c or c respecively. Corollary 13. Le α α ββ γρc C. Suppose ha Re(γ) > 0 and Re (ρ) > max {0 Re (α α β γ)re (α β )}. (4)

9 Mahemaical Problems in Engineering 9 Then he following formula holds: ρ 1 sin (c)) (x) = Γ(ρ)Γ(ρ +γ α α β)γ(ρ+β α ) Γ(ρ+β )Γ(ρ+γ α α )Γ(ρ+γ α β) xρ α α +γ 1 ρ ρ+1 ρ+γ α α β 3 ρ+β ρ+β +1 ρ+γ α α ρ+γ α α +β+1 ρ+γ α α +1 ρ+β α ρ+β α +1 ρ+γ α β ρ+γ α β+1 c x 4 ρ 1 sinh (c)) (x) = Γ(ρ)Γ(ρ +γ α α β)γ(ρ+β α ) Γ(ρ+β )Γ(ρ+γ α α )Γ(ρ+γ α β) xρ α α +γ 1 ρ ρ+1 ρ+γ α α β 3 ρ+β ρ+β +1 ρ+γ α α ρ+γ α α +β+1 ρ+γ α α +1 ρ+β α ρ+β α +1 ρ+γ α β ρ+γ α β+1 c x 4. (43) Corollary 14. Le α α ββ γρc C. Suppose ha Re(γ) > 0 and Re(ρ) < 1+min{Re( β) Re(α+α γ)re(α+ β γ)}.then ρ sin ( c )) (x) = Γ(α+α γ ρ)γ(α+β γ ρ) Γ( ρ)γ(α+α +β γ ρ) Γ( β ρ) Γ(α β ρ) xρ α α +γ α+α ρ 3 ρ ρ+1 ρ sinh ( c )) (x) α+α γ ρ+1 α+α +β γ ρ α+β γ ρ α+β γ ρ+1 α+α +β γ ρ+1 α β ρ β ρ β ρ+ 1 α β ρ+1 c 4x (44) = Γ(α+α γ ρ)γ(α+β γ ρ) Γ( ρ)γ(α+α +β γ ρ) Γ( β ρ) Γ(α β ρ) xρ α α +γ α+α ρ 3 ρ ρ+1 α+α γ ρ+1 α+α +β γ ρ α+β γ ρ α+β γ ρ+1 α+α +β γ ρ+1 β ρ β ρ+1 α β ρ α β ρ+1 c 4x.

10 10 Mahemaical Problems in Engineering 5. Concluding Observaions In his secion some consequences of he main resul derived in previous secions are given in deail. Also comparison wih oher known resuls from he lieraure is lised. (1) We remark ha all he resuls given by Purohi e al. 10 are followed from he resuls derived in his paper by seing b=1=c. () The resuls in Secions and 3 also provide he Marichev-Saigo-Maeda fracional inegraion of modified Bessel funcion and spherical Bessel funcions. (3) Se α = 0 in he operaors (1) and(). Then due o ideniies given by Saxena and Saigo 11 P.93i follows ha (I α0ββ γ f) (x) =(I γα γ β 0x f) (x) (I α0ββ γ f) (x) =(I γα γ β x f) (x). (45) The generalized inegral ransforms ha appear in he righ-hand side of he above equaions are due o Saigo 19 and defined as follows: (I αβη 0 + f) (x) = x α β x Γ (α) (x ) α 1 F 1 (I αβη f) (x) = 1 Γ (α) x 0 (α + β η α 1 )f() d x ( x) α 1 α β F 1 (α + β η α 1 x )f() d (46) where Γ(α) is he Euler gamma funcion 0 and F 1 (abcx)is he Gauss hypergeomeric funcion. The above fac helps us o conclude ha all he resuls givenin9 1 can also be obained from he resuls in his paper by seing α =0. (4) Noe ha Riemann-Liouville and Weyl and Erdélyi- Kober fracional calculus are special case of Saigo s operaor (46). Thus his paper is also useful o derive cerain composiion formula involving Riemann-Liouville and Weyl and Erdléyi- Kober fracional calculus and Bessel modified Bessel and spherical Bessel funcion of he firs kind. Conflic of Ineress The auhors declare ha here is no conflic of ineress regarding he publicaion of his paper. Acknowledgmens The research work presened here was suppored in par by he Deanship of Scienific Research (DSR) a Salman bin Abdulaziz Universiy for K. S. Nisar. References 1 O. I. Marichev Volerra equaion of Mellin convoluion ype wih a Horn funcion in he kernel Izvesiya Akademii Nauk BSSR. Seriya Fiziko-Maemaicheskikh Naukvol.1pp (Russian). M. Saigo and N. Maeda More generalizaion of fracional calculus in Transform Mehods & Special Funcions Varna 96 pp Bulgarian Academy of Sciences Bulgaria Sofia Y.C.KimK.S.LeeandH.M.Srivasava Someapplicaions of fracional inegral operaors and Ruscheweyh derivaives Mahemaical Analysis and Applicaions vol.197no. pp V. Kiryakova All he special funcions are fracional differinegrals of elemenary funcions Physics A: Mahemaical and Generalvol.30no.14pp K.S.MillerandB.RossAn Inroducion o he Fracional Calculus and Fracional Differenial Equaions JohnWiley& Sons New York NY USA H. M. Srivasava S.-D. Lin and P.-Y. Wang Some fracionalcalculus resuls for he H-funcion associaed wih a class of Feynman inegrals Russian Mahemaical Physics vol. 13 no. 1 pp Á. Baricz Geomeric properies of generalized Bessel funcions Publicaiones Mahemaicaevol.73no.1-pp S. R. Mondal and A. Swaminahan Geomeric properies of generalized Bessel funcions Bullein of he Malaysian Mahemaical Sciences Socieyvol.35no.1pp A. A. Kilbas and N. Sebasian Generalized fracional inegraion of Bessel funcion of he firs kind Inegral Transforms and Special Funcionsvol.19no.11-1pp S. D. Purohi D. L. Suhar and S. L. Kalla Marichev-Saigo- Maeda fracional inegraion operaors of he Bessel funcions Le Maemaichevol.67no.1pp R. K. Saxena and M. Saigo Generalized fracional calculus of he H-funcion associaed wih he Appell funcion F 3 Journal of Fracional Calculusvol.19pp C. Fox The asympoic expansion of generalized hypergeomeric funcions Proceedings of he London Mahemaical Sociey vol. s-7 no. 1 pp E. M. Wrigh The asympoic expansion of inegral funcions defined by Taylor series Philosophical Transacions of he Royal Sociey of London A: Mahemaical and Physical Sciences vol. 38 pp E. M. Wrigh The asympoic expansion of he generalized hypergeomeric funcion Proceedings of he London Mahemaical Socieyvol.46pp A. A. Kilbas M. Saigo and J. J. Trujillo On he generalized Wrigh funcion Fracional Calculus & Applied Analysis vol. 5 no. 4 pp A. A. Kilbas and N. Sebasian Fracional inegraion of he produc of Bessel funcions of he firs kind Fracional Calculus &AppliedAnalysisvol.13no.pp H. M. Srivasava Some Fox-Wrigh generalized hypergeomeric funcions and associaed families of convoluion operaors Applicable Analysis and Discree Mahemaics vol.1no.1pp A. Erdélyi W. Magnus F. Oberheinger and F. G. Tricomi Higher Transcendenal Funcions vol. I II McGraw-Hill Book Company New York NY USA 1953.

11 Mahemaical Problems in Engineering M. Saigo A remark on inegral operaors involving he Gauss hypergeomeric funcions Mahemaical Repors of College of General Educaion vol. 11 no. pp E. D. Rainville Special Funcions The Macmillan New York NY USA P. Malik S. R. Mondal and A. Swaminahan Fracional inegraion of generalized Bessel funcion of he firs kind in Proceedings of he ASME Inernaional Design Engineering Technical Conferences & Compuers and Informaion in Engineering Conference (IDETC/CIE 11) Washingon Wash USA Augus 011. S. G. Samko A. A. Kilbas and O. I. Marichev Fracional Inegrals and Derivaives Gordon and Breach Science Publishers Yverdon 1993 Translaed from he 1987 Russian original.

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