On Some Fractional Integral Operators Involving Generalized Gauss Hypergeometric Functions

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1 Available at Appl. Appl. Math. ISSN: Vol. 5, Issue (Decembe ), pp (Peviously, Vol. 5, Issue, pp ) Applicatios ad Applied Mathematics: A Iteatioal Joual (AAM) O Some Factioal Itegal Opeatos Ivolvig Geealized Gauss Hypegeometic Fuctios N. Vicheko ad O. Lisetska Depatmet of Mathematics ad Physics Natioal Techical Uivesity of Ukaie KPI Kyiv, Ukaie vicheko@hotmail.com S. L. Kalla * Istitute of Mathematics Vyas Istitute of Highe Educatio Jodhpu-348, Idia shyamkalla@yahoo.com Received: Febuay 3, ; Accepted: July, Abstact The object of this pape is to give a geealizatio of Gauss hypegeometic fuctio, ad to ivestigate its basic popeties. Futhe, we defie some factioal itegal opeatos ad thei iveses i tems of the Melli tasfom. Seveal well kow itegal opeatos, icludig Saigo opeatos ca be deived fom the esults established hee. Keywods: Factioal opeatos; Hypegeometic fuctios; Melli tasfom MSC No.: 6A33, 33C5. Geealized Gauss Hypegeometic Fuctio The impotat ole played by special fuctios, paticulaly the hypegeometic fuctio i solvig umeous poblems of Mathematical Physics, Egieeig ad Applied Mathematics is well kow [see, fo istace, Kilbas et al. (4), Kiyakova (994), Mathai et al. (973), * Coespodig Autho 3

2 3 N. Vicheko et al. Samko et al. (993) ad Vicheko et al. ()]. This fact has ispied the ivestigatio of seveal geealizatios of hypegeometic fuctios. With this objective i mid, we itoduce a geealizatio of the Gauss s hypegeometic fuctio i the followig fom: b cb a, F abcz, ; ; F t t zt ; ; dt bc, b t t, () whee RecReb, ;, z ; Re Re,,,, is the beta fuctio [Abamowitz et al. (97)]. ; ; deotes the geealized cofluet hypegeometic (itoduced by Vicheko (6)) as follows:, The fuctio acz c a c a, ; a ca c, () c; acz ; ; t t zt dt whee z is the paticula case of the so-called geealized Wight hypegeometic fuctio p q z (itoduced by Wight (935)) defied fo comple a i, b j ad eal A i, B j ( i, p, j, q ) by the seies,,,, b, B,, b, B p p p i i i p q z q q q b j j j a A a A a A z. (3) B! I a seies of papes, Wight (935, 94) also ivestigates its asymptotic behavio fo the lage values of agumet z ude the coditio q p B Ai. (4) j j i Cetai basic popeties of poved that p qz wee also studied i Kilbas et al. (). I paticula, it was p q z is a etie fuctio of z, ude coditio (4). Futhe, fo, geealized Gauss hypegeometic fuctio give by () has the followig et fom

3 AAM: Ite. J., Vol. 5, Issue (Decembe ) [Peviously, Vol. 5, Issue, pp ] 33 b c b a F abcz, ; ; F t t zt ; ; dt bc, b t t. (5) I the et sectios, we pove cetai basic popeties (Theoems -6, below) fo the geealized Gauss s hypegeometic fuctio F z ad we also defie some factioal itegal opeatos ad thei iveses. Coespodig esults fo the Gauss hypegeometic fuctio ad itegal opeatos of Kalla & Saea (969) ca be easily deived fom geeal esults established hee.. Basic Popeties of F z As idicated above, this sectio cosists of si theoems which povide some popeties of this fuctio. The fist theoem deals with a seies epesetatio, while the secod gives us some diffeetial popeties. F z. Ude the valid coditios of eistece of the F z (see ()), the followig fomula holds tue, Theoem. The seies epesetatio of fuctio z F abcz, ; ; a, b c, b ; bc, b!, (6) whee a is the Pochhamme symbol ad, is the, defied by [see Vicheko (6)]: y,,, ;,, ; ; t t geealized beta-fuctio y t t dt, (7) hee Re, Re y,, ;, ad is give by ()., Theoem. The diffeetial elatios fo F z. With due egad to covegece, the fuctio F abcz, ; ; satisfies the followig diffeetial fomulas: d, ; ; a b dz c F abcz d dz F a, b; c; z, (8) a, ; ; a z F abcza z, ; ; F abcz, (9)

4 34 N. Vicheko et al. d z F abcz, ; ; a F a, bcz ; ; F abcz, ; ;. () dz Some ecuece fomulae, additio ad multiplicatio popeties ae give i the followig theoems: Theoem 3. The fuctioal elatios fo F z. Ude the coditios of eistece of the fuctio the et fuctioal elatios ae valid: F z, ; ;, ; ;, ; ; b F a b c z c b F a b c z c F a b c z, (), ; ;, ; ;, ; ; c F abcz c b F abc z b F ab c z. () Theoem 4. The additio ad multiplicatio theoem. If the coditios of eistece of the fuctio ae valid, the the followig fuctioal elatios hold F z c a b y F abc, ; ; y F ab, c ; ;!, (3) c a b y F abcy, ; ; F ab, c ; ;!. (4) The followig two theoems deals with itegal epesetatios ad tasfomatio fomulae fo. F z F, we have the followig itegal epesetatios of, ; ; Theoem 5. Itegal epesetatios of fuctio F z a a bc, b z z. Fo the valid coditios of eistece of the F abcz : b ac t t, t t cb F abcz, ; ; ; ; dt, (5) t z b cb si cos, 4 a F abcz, ; ; ; ; d, (6) bc, b z si si

5 AAM: Ite. J., Vol. 5, Issue (Decembe ) [Peviously, Vol. 5, Issue, pp ] 35 a c c a b sh ch ch ba 3, a 4 bc, b sh F abcz, ; ; ; ; z ch b z ch z cb acb sh ch 3, ch a d, (7) F abcz, ; ; ; ; d. (8) bc, b sh shθ ad chθ stad fo usual hypebolic sie ad cosie, espectively. Theoem 6. Tasfomatios fo F z. Ude the coditios of eistece of the fuctios F abcz, ; ; the followig geealized elatios of Kumme types ae valid:, ; ; a, ; ; F a b a b c z z F a a c a b c z, (9) z Fa c b c c z z Fa c b c z c a, ; ;, ; ;, (), ; ; ac, ; ; F c a c b c a b z z F c a a c a b z, () a z F abcz, ; ; z F ac, bc ; ;. () z Poofs. We obseve that the Theoems -3 ca be easily deived, usig defiitios () & (7), ad simple tasfomatios. To pove the esults (3) ad (4), we use fomula (8) of Theoem, ad the Taylo theoem about aalytical fuctios [Kilbas et al. (4)]. Poof of the Theoem 5: Usig mai itegal epesetatio () of, ; ; t si, F abcz ad substitutig y t, y t ch, ch t, we get espectively fomulae (5), (6), (7) ad (8). ch

6 36 N. Vicheko et al. Poof of the Theoem 6: Simple tasfomatios lead to esults (9) to (). Fo eample, let us pove (9) ad (). We make the followig tasfomatios: b ac a, F a, b; ab c;z t t zt ; ; dt ba, c t t b a c a, y y z y ; ; dy ba, c yy a z F a, ac; ab c;z. Similaly, we have b cb a, F abcz, ; ; t t zt ; ; dt bc, b t t b c b a, y y z y ; ; dy bc, b yy a z z F a, cb; c;. z This poves the esult (). 3. The Itegal Opeatos with F z. Riema-Liouville opeatos of factioal itegatio ad diffeetiatio ae basic i the theoy of factioal calculus. Seveal authos, like Love, Saea, Kalla & Saea (969), Saigo, McBide etc. [see, Kiyakova (994)] have defied diffeet foms of hypegeometic opeatos of factioal itegatio. Kalla (969) has defied itegal opeatos ivolvig Fo H-fuctio. I this pat of the pape, we defie some factioal itegal opeatos ivolvig the fuctio F z give by equatio (5) ad thei iveses i tems of the Melli tasfom. The coespodig esults fo hypegeometic opeatos metioed above ca be deived fom geeal fomulas established hee. Let us itoduce the followig fou itegal opeatos: a abc,, a c d t I f t F a, bc; ac; f t dt, (3) ac d

7 AAM: Ite. J., Vol. 5, Issue (Decembe ) [Peviously, Vol. 5, Issue, pp ] 37 whee a, c ; b ad d ae eal, the fuctio f is cotiuous o, ; ab, bd,,, ; ; B f t F a b c t f t dt, ( a, c ); (4) c d abd abcd,,, d a a t I f t t F abd, ac; a; f t dt,,, s ( a, d ; b, c ae eal); (5) f t t F, s; s; f t dt. (6) t It is iteestig to obseve that fo, the opeato give by (5) yields to the Saigo s factioal itegal opeato [see Kiyakova (994)]. The Melli ad Laplace tasfoms ae used to obtai the ivese opeatos. The opeatos defied by (3) to (6), give the followig espective ivese opeatos whee abc,, ab, a c, c abc,, I f K I M I f, c, (7) M. deotes the Melli itegal tasfom ad K c c a b c b a c bcab cac. (8) The ivese opeato fo (4) is as follows: whee, ab bd a Bc, d, f KD limlh, (9) h,

8 38 N. Vicheko et al. a p bdd p cdb a, b C U t t I MB c, d, f, d t, dt cbd, p p Uf f, d Lf f!! d, (,3, 4, ), K d d c b d b d a bc, b dcd. Now we coside the ivese opeato fo (5), abcd,,, abcd,,, abcd,,, I f K I M I f a, (3) 3, whee a d a t, ; ;, (3) abd abcd,,, I f t t F abd c a f t dt K 3 a a c c a a a c c a. Fially, the ivese opeato fo (6) is deived as follows: whee,, s 4,, s f K T V W VT M f, s, (3), 4 4 T f L t f t, t, (33)

9 AAM: Ite. J., Vol. 5, Issue (Decembe ) [Peviously, Vol. 5, Issue, pp ] 39 L is the ivese Laplace tasfom, W f W f ad,, W f, is the Edelyi-Kobe factioal itegal opeato, [see Kiyakova (994), pp.65-66] Vf f K 4, V f f, s s s s, s. The above ivesio fomulae ca be deived easily. The epesetatio of itegal opeato,, i the followig fom ca be used to pove the esult (3). whee s f K M T I T f s,, (34),,, 5 t T f t e f t dt, I, f t t f t dt, (35) ad M is the ivese Melli tasfom. Seveal special cases ca be deived fom these geeal esults. Ivesio fomula fo Saigo s opeato ca be deived fom (3). 4. Discussios We have give a geealizatio of Gauss hypegeometic fuctio ad deived some of its popeties. This fuctio is used to defie ew opeatos of factioal itegatio. Thei ivese opeatos ae obtaied by a appeal to the Laplace ad Melli tasfoms. Futhe, the geealized Gauss s hypegeometic fuctio ad its popeties established i this pape ae of geeal foms ad may be used to defie ew desity fuctios i pobability theoy. These opeatos of factioal itegatio ca be used to solve cetai factioal itegodiffeetial equatios.

10 33 N. Vicheko et al. Ackowledgemet The authos ae thakful to the efeees fo thei useful suggestios. REFERENCES Abamowitz, M., I. ad Stegu, I.A. (97). Hadbook of Mathematical Fuctios with, Fomulas, Gaphs ad Mathematical Tables. Dove, New Yok. Kalla, S.L. ad Saea, R.K. (969). Itegal opeatos ivolvig hypegeometic fuctios. Mathematische Zeitschift. Vol. 8, Kalla, S.L. (969), Itegal opeatos ivolvig Fo s H-fuctio. Acta Meicaa de Ciecias y Techologia, Vol. 3, 7-. Kilbas, A.A. ad Saigo, M. (4). H-Tasfoms, Chapma ad Hall/CRC Pess, Boca Rato, FL. Kilbas, A.A., Saigo, M ad Tujillo, J.J. (). O the geealized Wight fuctio, Factioal Calculus ad Applied Aalysis. Vol.5, Kiyakova, V. (994). Geealized Factioal Calculus ad Applicatios, Wiley & Sos. Ic., New Yok. Mathai, A. M., ad Saea, R.K. (973). Geealized Hypegeometic Fuctios With Applicatios i Statistics ad Physical Scieces, Spige-Velag, Beli. Samko, S.G., Kilbas, A.A. ad Maichev, O.I. (993). Factioal Itegals ad Deivatives- Theoy ad Applicatios, Godo ad Beach Sciece Publishes, New Yok. Vicheko, N. Kalla, S.L. ad Al Zamel, A. (). Some esults o a geealized hypegeometic fuctio, Itegal Tasfoms ad Special Fuctios. Vol., 89-. Vicheko, N. (6). O the geealized cofluet hypegeometic fuctio ad its applicatios. Factioal Calculus Applied Aalysis. Vol.9, -8. Wight, E.M. (935). The asymptotic epasio of the geealized hypegeometic fuctio. Joual of the Lodo Mathematical Society. Vol., Wight, E.M. (94). The asymptotic epasio of the geealized hypegeometic fuctio. Joual of the Lodo Mathematical Society ().Vol. 46,