Research Article Certain Integral Transform and Fractional Integral Formulas for the Generalized Gauss Hypergeometric Functions

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1 Hindawi Publishing Cororaion Absrac and Alied Analysis Volume 24, Aricle ID , 7 ages h://d.doi.org/.55/24/ Research Aricle Cerain Inegral Transform and Fracional Inegral Formulas for he Generalized Gauss Hyergeomeric Funcions Junesang Choi and Praveen Agarwal 2 Dearmen of Mahemaics, Dongguk Universiy, Gyeongju 78-74, Reublic of Korea 2 DearmenofMahemaics,AnandInernaionalCollegeofEngineering,Jaiur,NearKanoa,AgraRoad,Bassi,Jaiur332,India Corresondence should be addressed o Junesang Choi; junesang@mail.dongguk.ac.kr Received 25 Aril 24; Revised 23 May 24; Acceed 4 June 24; Published 6 June 24 Academic Edior: Soheil Salahshour Coyrigh 24 J. Choi and P. Agarwal. This is an oen access aricle disribued under he Creaive Commons Aribuion License, which ermis unresriced use, disribuion, and reroducion in any medium, rovided he original work is roerly cied. A remarkably large number of inegral ransforms and fracional inegral formulas involving various secial funcions have been invesigaed by many auhors. Very recenly, Agarwal gave some inegral ransforms and fracional inegral formulas involving he F (α,β) ( ). In his sequel, using he same echnique, we esablish cerain inegral ransforms and fracional inegral formulas for he generalized Gauss hyergeomeric funcions F (α,β,m) ( ). Some ineresing secial cases of our main resuls are also considered.. Inroducion and Preliminaries The heory of secial funcions has been one of he mos raidly growing research subjecs in mahemaical analysis. A lo of secial funcions of mahemaical hysics and engineering, such as Jacobi and Laguerre olynomials, can be eressed in erms of he generalized hyergeomeric funcions or confluen hyergeomeric funcions (see, e.g.,, ages 66 69). Therefore, he corresonding eensions of several oher familiar secial funcions are eeced o be useful and need o be invesigaed (see, e.g., 2 7 and,fora very recen work, see also 8). Very recenly, Parmar 9 inroduced and invesigaed some fundamenal roeries and characerisics of he more generalized bea ye funcion (, y) defined by (see 9,age37,Equaion(9)) Iiseasyoseehahesecialcaseof()whenmreduces o he well-known generalized bea ye funcion defined by (see, e.g., 7, age 462, Equaion (4); see also 6, age32, Chaer 4) B (α,β) (, y) (B (α,β;) (, y)) ( ) y F (α; β; ( ) )d (R () > ; min {R (), R (y), R (α), R (β)} > ). (2) For αβ,(2) reduces o he eended bea ye funcion, due o Chaudhry e al. 4,age59,Equaion(.7)(seealso 3,age2,Equaion(.7)), defined by (, y) : ( ) y F (α; β; m m )d ( ) (R () > ; min {R (), R (y), R (α), R (β)}>, R (m) >). () B (, y) (B (α,α) (, y)) ( ) y e ( ( ) )d (R () > ). (3)

2 2 Absrac and Alied Analysis The classical Euler s bea funcion B(, y) is defined by (see, e.g.,, ages 7 ) B(,y): ( ) y d (R () >,R (y) > ). (4) I is easy o see he following relaion: B(,y):B (, y) B (α,β) (, y) B (α,β;) (, y). (5) By making use of B (, y), Chaudhryeal.4, age59, Equaions (2.) and (2.2) eended he Gauss s hyergeomericfuncionasfollows: F (a, b; c; z) : n B z n n! ( z <), (6) where R(c) > R(b) > and R() and denoes he Pochhammer symbol given in (). Similarly, by aealing o B (α,α) (, y), Ozergin e al. inroduced and invesigaed a furher eension of he following oenially useful generalized Gauss hyergeomeric funcions defined as follows (see, e.g., 7, age466, Secion3; see also 6,age39,Chaer4): F (α,β) B (α,β) z n (a, b; c; z) ( z <), n B (b, c b) n! (7) where min{r(α), R(β)} >, R(c) > R(b) >,andr(). By using he more generalized bea funcion (), Parmar 9 inroduced and invesigaed a family of he following oenially useful generalized Gauss hyergeomeric funcions defined as follows (see 9,age44): (a, b; c; z) n ( z <), where min{r(α), R(β), R(m)} > ; R(c) > R(b) > and R(). I is obvious o see ha F (α,β;) (a, b; c; z) F (α,β) (a, b; c; z), F (α,α;) (a, b; c; z) F (a, b; c; z), F (α,α;) (a, b; c; z) 2 F (a, b; c; z), where he 2 F ( ) is a secial case of he well-known generalized hyergeomeric series F q ( ) defined by (see, e.g.,, Secion.5; see also ) α,...,α ; (α F q z ) n (α ) n z n β,...,β q ; n (β ) n (β q ) n! n F q (α,...,α ;β,...,β q ; z), () z n n! (8) (9) where (λ) n is he Pochhammer symbol defined (for λ C)by (see, age 2 and ages 4 6) (n ) (λ) n : { λ (λ+),...,(λ+n ) (n N : {, 2, 3,...}) Γ (λ+n) Γ (λ) (λ C \ Z ). () and heir secial cases o many diverse areas of mahemaical, hysical, engineering, and saisical sciences (see, for deails, 9 and he references cied herein). Very recenly, Agarwal gave some ineresing inegral ransform and fracional inegral formulas involving (7).Inhissequel,usinghesameechnique,werooseo derive some inegral ransforms and image formulas for he generalized Gauss hyergeomeric funcion (8) by alying cerain inegral ransforms (like bea ransform, Lalace ransform, and Whiaker ransforms) and general air of fracional inegral oeraors involving Gauss hyergeomeric funcion 2 F, which will be inroduced in Secions 2 and 3, resecively.wealsoconsidersomeineresingsecialcases of our main resuls. Here, Γ is he familiar Gamma funcion and C and Z denoe ses of comle numbers and nonosiive inegers, resecively. The above-menioned deailed and sysemaic invesigaion was indeed moivaed largely by a demonsraed oenial for alicaions of he more generalized Gauss hyergeomeric funcion 2. Inegral Transform and he Generalized Gauss Hyergeomeric Funcions In his secion, we will rove hree heorems, which ehibi he connecions beween he Euler, Lalace, and Whiaker inegral ransforms and he generalized Gauss hyergeomeric ye funcions (a,b;c;z)defined by (8). Theorem. Suose ha R(), min{r(α), R(β), R(m)} >, R(c) > R(b) >, andl C are arameers. Then, he following bea ransform formula holds: B{ (l+m,b;c;yz):l,m} B(l, m) n (l) n B(l, m) (l,b;c;y) ( y <), y n n! where he bea ransform of f(z) is defined as (see 2) (2) B {f (z) :a,b} z a ( z) b f (z) dz. (3) Furher, i is assumed ha he involved Euler (bea) ransforms eis.

3 Absrac and Alied Analysis 3 Proof. Using (3) and alying (8) o he Euler(bea) ransform of (2), we ge z l ( z) m (l+m,b;c;yz)dz z l ( z) m n (l+m) n (b+ n,c b) (yz) n dz. B (b, c b) n! (4) By changing he order of inegraion and summaion and using bea inegral, we obain z l ( z) m (l+m,b;c;yz)dz n (l+m) n Γ (l) Γ (m) Γ (l+m) n (l) n Γ (l+n) Γ (m) (y) n Γ (l+m+n) n! (y) n, n! which, uon using (8), yields our desired resul (2). (5) Theorem 2. Suose ha y, R(s) >, min{r(α), R(β), R(m)} >, R(c) > R(b) >, R(),and y/s <.Then, he following Lalace ransform formula holds: L {z l (a, b; c; yz)} Γ (l) s l (a, l, b; c; y s ), (6) where he Lalace ransform of f(z) is defined as (see 2) L{f(z)} e sz f (z) dz, (7) and assume ha boh sides of (6) eis. Proof. Using (7) and alying (8), we ge z l e sz (a,b;c;yz)dz z l e sz n (yz) n dz. n! (8) By changing he order of inegraion and summaion and using Lalace ransform, we ge z l e sz (a,b;c;yz)dz n Γ (l+n) s l+n n (y), n! which, uon using (8), yields our desired resul (6). (9) Theorem 3. Suose ha w, R(), ρ and δ C. Then, he following Whiaker ransform formula holds: ρ e δ/2 W λ,μ (δ) (a, b; c; w) d δ ρ Γ((/2) +μ+ρ)γ((/2) μ+ρ) Γ( λ+ρ) 2, (a, 2 +μ +ρ, 2 μ+ρ,b;c, λ+ρ;w δ ), (2) rovided ha he inegral involving Whiaker ransform converges. Proof. Alying (8)andseingδ in he lef-hand side of (2), we ge ( δ )ρ e /2 W λ,μ () n (w) n δ n δn! d. (2) By changing he order of inegraion and summaion, we obain δ ρ n w n δ n n! ρ+n e /2 W λ,μ () d. (22) Ne, we can use he following inegral formula involving he Whiaker funcion: e /2 W λ,μ () d Γ((/2) +μ+)γ((/2) μ+) Γ ((/2) λ+) (R ( ±μ)> 2 ). (23) Then, afer a lile simlificaion, (22)becomeshefollowing form: ρ e δ/2 W λ,μ (δ) (a, b; c; w) d δ ρ n w n δ n n! Γ((/2) +μ+ρ+n)γ((/2) μ+ρ+n), Γ( λ+ρ+n) which, uon using (8), yields our desired resul (2). (24) I may be noed in assing ha he secial cases of Theorems o 3 when m immediaely reduce o he corresonding resuls due o Agarwal.

4 4 Absrac and Alied Analysis 3. Fracional Calculus of he Generalized Gauss Hyergeomeric Funcions Recenly, fracional inegral oeraors involving he various secial funcions have been invesigaed by several auhors (see, e.g.,, 3 26; see also 27, 28). Here, in his secion, we will esablish some fracional inegral formulas for he generalized Gauss hyergeomeric ye funcions (a,b;c;z). To do his, we need o recall he following air of Saigo hyergeomeric fracional inegral oeraors. For >, μ,,η C and R(α) >,wehave (I μ,,η, μ f ()) () Γ(μ) ( ) μ 2 F (μ +, η; μ; )f() d, (25) (J μ,,η, f ()) () Γ(μ) ( ) μ μ 2 F (μ +, η; μ; )f() d, (26) where he 2 F ( ), asecialcaseofhegeneralizedhyergeomeric funcion (), is he Gauss hyergeomeric funcion. The oeraor I μ,,η, ( ) conains boh he Riemann- Liouville R μ, ( ) and he Erdélyi-Kober Eμ,η, ( ) fracional inegral oeraors by means of he following relaionshis: (R μ, (E μ,η, f ()) () (Iμ, μ,η, f ()) () Γ(μ) ( ) μ f () d, f ()) () (Iμ,,ηf ()) (), μ η Γ(μ) ( ) μ η f () d. (27) (28) I is noed ha he oeraor (26) unifies he Weyl ye and he Erdélyi-Kober fracional inegral oeraors as follows: We also use he following image formulas which are easy consequences of he oeraors (25) and(26)(see24, 26): (I μ,,η Γ (λ) Γ(λ +η), λ ) () Γ (λ ) Γ(λ+μ+η) λ (λ >, λ +η>), (J μ,,η Γ ( λ+) Γ(η λ+), λ ) () Γ ( λ) Γ( +μ λ+η+) λ (3) (β λ+>,η λ+>). (32) The Saigo fracional inegraions of generalized Gauss hyergeomeric ye funcions (8) are given by he following resuls. Theorem 4. Le >, R(), andμ,,η,ρ,e C be arameers such ha R (μ) >, R (ρ) >, R (ρ) > ma {, R ( η)}. (33) Then, he following fracional inegral formula holds: (I μ,,η, ρ (a, b; c; e)) () ρ 2,2 Γ(ρ)Γ(ρ +η) Γ(ρ+μ+η)Γ(ρ ) a, b, ρ, ρ +η; e. c, ρ,ρ+μ+η; (34) Proof. For convenience, we denoe he lef-hand side of he resul (34) by I. Using(8), and hen changing he order of inegraion and summaion, which is valid under he condiions of Theorem 4,we find I (I μ,,η, n ρ a n n a n (e) n n! ) () e n n! (Iμ,,η, { ρ+n }) (). (35) (W μ, (K μ,η, f ()) () (Jμ, μ,ηf ()) (), Γ(μ) ( ) μ f () d, f ()) () (Jμ,,ηf ()) () η, Γ(μ) ( ) μ μ η f () d. (29) (3) Now, making use of he resul (3), we obain I ρ a n n Γ(ρ+n)Γ(ρ +η+n) (e) n. Γ(ρ +n)γ(ρ+μ+η+n) n! This, in view of (8), gives he desired resul (34). (36)

5 Absrac and Alied Analysis 5 Theorem 5. Le >, R(), andμ,,η,ρ,e C be arameers saisfying he following inequaliies: Corollary 8. Le >, R(),andμ, ρ C be arameers such ha R(μ) > and R(ρ) >.Then,oneges R (μ)>, R (ρ)>, R (ρ) <+min {R (η), R ()}. (37) (R μ, ρ (a, b; c; e)) () Then, he following fracional inegral formula holds: (J μ,,η, ρ (a, b; c; e )) () ρ+μ Γ(ρ) a, b, ρ; Γ(ρ+μ), c, ρ + μ ; e. (4) ρ Γ( ρ+)γ( ρ+η) Γ( ρ)γ( ρ η+ +μ) a, b, ρ +, ρ+η; 2,2 c, ρ, ρ+μ+ η; e. (38) Proof. As in he roof of Theorem 4,akingheoeraor(26) and he resul (32) ino accoun, one can easily rove (38). Therefore, we omi he deails of he roof. Seing in Theorems 4 and 5 and emloying he relaions (28) and(3) yield cerain ineresing resuls assered by he following corollaries. Corollary 6. Le >, R(), andμ, η, ρ, e C be arameers such ha R(μ) >, R(ρ) >, and R(ρ) > R( η). Then,herigh-sideErdélyi-Kober fracional inegrals of he generalized Gauss hyergeomeric ye funcions are given by (E μ,η, ρ ρ Γ(ρ η) Γ(ρ+μ+η), (a, b; c; e)) () a, b, ρ η; e. c, ρ + μ + η; (39) Corollary 7. Le >, R(), andμ, η, ρ, e C be arameers saisfying he following inequaliies R(μ) >, R(ρ) >,andr(ρ) < + R(η).Then,onehas (K μ,η, ρ ρ Γ( ρ+η) Γ( ρ η+μ) (a, b; c; e )) () a, b, ρ + η;, c, ρ+μ η; e. (4) Furher, if we relace wih μ in Theorems 4 and 5 andmakeuseofherelaions(27) and(29), we obain oher Riemann-Liouville and Weyl fracional inegrals of he generalized Gauss hyergeomeric ye funcion given by he following corollaries. Corollary 9. Le >, R(),andμ, ρ C be arameers saisfying he following inequaliies R(μ) > and R(ρ) >. Then, one obains (W μ, ρ (a, b; c; e )) () ρ+μ Γ( ρ μ) Γ( ρ), a, b, ρ μ; c, ρ ; e. (42) 4. Concluding Remarks We can also resen a large number of secial cases of our main fracional inegral formulas in Theorems 4 and 5. Here, we illusrae wo more formulas. Seing α β in (34) and(38), resecively, and using he known formula due o Lee e al. (see, 5, age 97, Equaion (6.)), we obain cerain ineresing (resumably) new fracional inegral formulas involving he eended hyergeomeric funcion F (m) (a,b;c;z)assered by he following corollaries. Corollary. Le >, R(),andμ,,η,ρ,e C be arameers such ha R (μ) >, R (ρ) >, R (ρ) > ma {, R ( η)}. (43) Then, one has (I μ,,η, ρ F m (a, b; c; e)) () ρ Γ(ρ)Γ(ρ +η) Γ(ρ+μ+η)Γ(ρ ) 2 F m a, b, ρ, ρ +η;,2 e. c, ρ,ρ+μ+η; (44) Corollary. Le >, R(), andμ,,η,e,ρ C be arameers saisfying he following inequaliies: R (μ)>, R (ρ)>, R (ρ)< +min {R (η), R ()}. (45)

6 6 Absrac and Alied Analysis Then, one ges (J μ,,η, ρ F (a,b;c; e )) () ρ β Γ( ρ+)γ( ρ+η) Γ( ρ)γ( ρ η+ +μ) (46) a, b, ρ +, ρ+η; e 2 F,2. c, ρ, ρ+μ+ η; Seing mand αβin Theorems 4 and 5 and making use of he relaion (6) yield some known fracional inegral formulas due o Agarwal. Also seing mand making use of he relaion (7) give he known inegral ransforms and fracional inegral formulas due o Agarwal. Furher, if we se m and in Theorems o 5 or make use of he resul (8), we obain various inegral ransforms and fracional inegral formulas for he Gauss hyergeomeric funcion 2 F. The generalized Gauss hyergeomeric ye funcions defined by (8) ossess he advanage ha mos of he known and widely invesigaed secial funcions are eressible in erms of he generalized Gauss hyergeomeric funcions (see 9). We may also emhasize ha resuls derived in his aer are of general characer and can secialize o give furher ineresing and oenially useful formulas involving inegral ransform and fracional calculus. Finally, i is eeced ha he resuls resened here wih oenial secial cases can find some alicaions in robabiliy heory and o he soluions of fracional differenial and inegral equaions. Conflic of Ineress The auhors declare ha here is no conflic of ineress regarding he ublicaion of his aer. Acknowledgmens The auhors would like o eress heir dee graiude for he reviewers helful commens. This research was, in ar, suored by he Basic Science Research Program hroughhe Naional Research Foundaion of Korea funded by he Minisry of Educaion, Science and Technology of he Reublic of Korea (Gran no. 2-5). This work was suored by Dongguk Universiy Research Fund. References H. M. Srivasava and J. Choi, Zea and q-zea Funcions and Associaed Series and Inegrals, Elsevier Science, Amserdam, The Neherlands, P. Agarwal, M. Chand, and S. D. Purohi, A noe on generaing funcions involving generalized Gauss hyergeomeric funcions, Naional Academy Science Leers.Inress. 3M.A.Chaudhry,A.Qadir,M.Rafique,andS.M.Zubair, Eension of Euler's bea funcion, Comuaional and Alied Mahemaics,vol.78,no.,.9 32,997. 4M.A.Chaudhry,A.Qadir,H.M.Srivasava,andR.B.Paris, Eended hyergeomeric and confluen hyergeomeric funcions, Alied Mahemaics and Comuaion,vol.59,no.2, , D. M. Lee, A. K. Rahie, R. K. Parmar, and Y. S. Kim, Generalizaion of eended bea funcion, hyergeomeric and confluen hyergeomeric funcions, Honam Mahemaical Journal,vol.33,no.2,.87 26,2. 6 E. Özergin, Some roeries of hyergeomeric funcions Ph.D. hesis, Easern Medierranean Universiy, Gazimağusa, Norh Cyrus, 2. 7 E. Özergin, M. A. Özarslan, and A. Alın, Eension of gamma, bea and hyergeomeric funcions, Comuaional and Alied Mahemaics,vol.235,no.6,.46 46,2. 8 H. Liu and W. Wang, Some generaing relaions for eended Aell's and Lauricella's hyergeomeric funcions, The Rocky Mounain Mahemaics.Inress. 9 R. K. Parmar, A new generalizaion of gamma, bea, hyergeomeric and confluen hyergeomeric funcions, Le Maemaiche,vol.68,no.2,.33 42,23. H. M. Srivasava and P. W. Karlsson, Mulile Gaussian Hyergeomeric Series, Halsed Press, Ellis Horwood Limied, Chicheser, UK, John Wiley and Sons, New York, NY, USA, 985. P. Agarwal, Cerain roeries of he generalized Gauss hyergeomeric funcions, Alied Mahemaics & Informaion Sciences,vol.8,no.5, ,24. 2 I. N. Sneddon, The Use of Inegral Transforms, TaaMcGraw- Hill, New Delhi, India, P. Agarwal, Fracional inegraion of he roduc of wo mulivariables H-funcion and a general class of olynomials, in Advances in Alied Mahemaics and Aroimaion Theory, vol. 4 of Sringer Proceedings in Mahemaics & Saisics, , Sringer, New York, NY, USA, P. Agarwal, Furher resuls on fracional calculus of Saigo oeraors, Alicaions and Alied Mahemaics, vol. 7, no. 2, , P. Agarwal, Generalized fracional inegraion of he H funcion, Le Maemaiche,vol.67,no.2,.7 8,22. 6 P. Agarwal and S. Jain, Furher resuls on fracional calculus of Srivasava olynomials, Bullein of Mahemaical Analysis and Alicaions,vol.3,no.2,.67 74,2. 7 P. Agarwal and S. D. Purohi, The unified ahway fracional inegral formulae, Fracional Calculus and Alicaions,vol.4,no.9,. 8,23. 8 A. A. Kilbas, Fracional calculus of he generalized Wrigh funcion, Fracional Calculus & Alied Analysis, vol.8,no.2,. 3 26, A. A. Kilbas, H. M. Srivasava, and J. J. Trujillo, Theory and Alicaions of Fracional Differenial Equaions, vol.24of Norh-Holland Mahemaical Sudies, Elsevier Science, Amserdam, The Neherlands, V. Kiryakova, On wo Saigo's fracional inegral oeraors in he class of univalen funcions, Fracional Calculus & Alied Analysis,vol.9,no.2,.6 76,26. 2 V. Kiryakova, A brief sory abou he oeraors of he generalized fracional calculus, Fracional Calculus & Alied Analysis, vol., no. 2, , S. Kumar, D. Kumar, S. Abbasbandy, and M. M. Rashidi, Analyical soluion of fracional Navier-Sokes equaion byusing modifed Lalace decomosiion mehod, Ain Shams Engineering Journal,vol.5, ,24.

7 Absrac and Alied Analysis 7 23 M. Saigo, On generalized fracional calculus oeraors, in Proceedings of he Inernaional Worksho on Recen Advances in Alied Mahemaics, , Kuwai Universiy, Kuwai, M. Saigo, A remark on inegral oeraors involving he Gauss hyergeomeric funcions, Mahemaical Reors of College of General Educaion. Kyushu Universiy, vol., no. 2, , M. Saigo, A cerain boundary value roblem for he Euler- Darbou equaion I, Mahemaica Jaonica, vol.24,no.4, , M. Saigo and N. Maeda, More generalizaion of fracional calculus, in Proceedings of he 2nd Inernaional Worksho on Transform Meods and Secial Funcions,P.Rusev,I.Dimovski, and V. Kiryakova, Eds., , IMI, Varna, Bulgaria, Augus H. M. Srivasava and P. Agarwal, Cerain fracional inegral oeraors and he generalized incomlee hyergeomeric funcions, Alicaions and Alied Mahemaics, vol.8,no.2, , A.-M. Yang, Y.-Z. Zhang, C. Caani e al., Alicaion of local fracional series eansion mehod o solve Klein-Gordon equaions on Canor ses, Absrac and Alied Analysis, vol. 24,AricleID37274,6ages,24.

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