FRACTIONAL CALCULUS OF GENERALIZED K-MITTAG-LEFFLER FUNCTION

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1 Joual of Rajastha Academy of Physical Scieces ISSN : ; URL : htt://aos.og.i Vol.5, No.&2, Mach-Jue, 26, FRACTIONAL CALCULUS OF GENERALIZED K-MITTAG-LEFFLER FUNCTION Jiteda Daiya ad Jeta Ram Deatmet of Mathematics ad Statistics, Jai Naai Vyas Uivesity, Jodhu s: jiteda.daiya25@gmail.com, bishoi_j@yahoo.com Abstact: This ae deals with the deivatio of the -factioal diffeetiatio ad -factioal itegatio of the geealized -Mittag-Leffle fuctio defied ad studied by Saxea et al. [3]. The esults deived i this ae ovide extesio of the esults give by Kilbas et al. [4,5], Saxea [2], Saxea ad Saigo [] ad Saxea et al []. The esults obtaied ae useful i alied oblems of sciece, egieeig ad techology. Key wods: Geealized -Mittag-Leffle fuctio, -factioal diffeetiatio, -factioal itegatio, -Pochhamme symbol, beta fuctio. Mathematics Subject Classificatio: 26A33, 33C6. Itoductio The -factioal itegal is defied ad studied by Mubee ad Habibullah [7] i the fom: x I [ f ( x] ( x t f ( t dt. Re(. ( The -factioal diffeetiatio is defied by Romeo et al. [9] i the fom: d D f ( t I f ( t, R ad. (2 dt O x d d D ( ( ( f t I f t x t f ( t dt. Re(. dt dt ( ad ( (3

2 9 Jiteda Daiya ad Jeta Ram D f ( t I f ( t ( x t f ( t dt. x d d dt dt ( (4 The -Pochhamme symbol has bee itoduced i [] i the fom: ( x x( x ( x 2... ( x (,, ( x ( q, ( x q, ( x q q,, (5 whee xc; R ad N. Poositio Let C ad, s R, the the followig idetity holds: s s s(, s ad i aticula (, Poositio 2. Let C;, s R ad N, the the followig idetity holds: (6 (7 q s ( q, s, s ad i aticula q ( q,, q q, Note : Fo futhe details of -Pochamme symbol, -secial fuctios ad factioal Fouie tasfom oe ca efe to the aes by Romeo et al [8] ad Mubee ad Habibullah [9]. Defiitio : Let R;,, C; Re( ad C, the the geealized -Mittag- Leffle fuctio is give by Saxea et al. [3] E,,, ( z (, z (! ( whee ( x,( x, C deotes the Pochhamme symbol with (! fo N N {}, (8 (9

3 Factioal Calculus of... 9 which is defied i tems of gamma fuctio as (also see [6] ( x ( x ( x ( ; xc\{} x( x... ( x ( N; xc Secial cases of E,,, ( z (i (ii (ii Fo = q, equatio (7 yields geealized K-Mittag-Leffle fuctio defied by Saxea et al [2] ( E z E z, q z,,,,, (! q, q ( Fo =, equatio ( yields geealized Mittag-Leffle fuctio defied by Shula ad Pajaati [5] ( E z E z, q z,,,, (! q q (2 Whe q =, equatio ( gives the Mittag-Leffle fuctio defied by Dooego & Ceutti [2]., ( z,,,, (!, (3 E z E z Note 2: A detailed accout of Mittag-Leffle fuctio ad thei alicatio ca be foud i the suvey aes by Haubold et al. [3], Mathai et al. [6], Saxea [2] ad Saxea et al [4]. Theoem. If R ;,, C ;Re(, C ad N N {} the / /, / / /, /,,,, I t E ( wt ( x x E ( w x Poof : Usig equatio ( ad (, it gives x /, / /, /,,,, ( I t E ( wt ( x ( x t t E ( wt dt x / / (, ( wt ( x t t dt ( (! (4

4 92 Jiteda Daiya ad Jeta Ram ( ( ( (! We set t x, w / / xu ( ( ( (! ( x t t dt, w / / ( x xu ( xu dt / / / (, ( wx / / / x ( (! ( u u du / / ( / x, ( wx ( (! By vitue of the elatio (7, the above exessio becomes / / ( / x, ( w x ( / /! / / / / (, ( w x x This comletes the oof of Theoem. Coollay. Fo equatio (4 educe i the Riema-Liouville factioal itegal! / /, / /, /,,,, I t E ( wt ( x x E ( w x Coollay.2 Fo q, equatio (4 educes i the followig fom / /, q / / /, q /,,,, I t E ( wt ( x x E ( w x (5 (6

5 Factioal Calculus of Coollay.3 Whe equatio (6 gives /, q x, q, /, I t E ( wt ( x E ( w x Coollay.4 Whe q ad equatio (4 yields I t E, ( wt ( x x E, ( w x Theoem 2. Let R ;,, C ;Re(, C ad N N {} the, we have /, / / /, /,,,, D t E ( wt ( x x E ( w x Poof : I view of (4 ad ( gives /, / d /, /,,,, D t E ( wt ( x I t E ( wt ( x dx x d /, / ( x t t E,, ( wt dt dx ( x / / (, ( wt d ( x t t dt dx ( (! d ( ( dx ( (! We set t xu x (, w / / d ( ( dx ( (! ( x t t dt x (, w / / ( x xu ( xu dt ( / / ( / (, ( / / d x w dx ( (! ( u u du (7 (8 (9

6 94 Jiteda Daiya ad Jeta Ram ( / / ( / ( d x, ( w dx ( ( (! Usig eq. (9, the above exessio becomes / / ( / (, ( ( / (! d x w dx ( / / ( / (, ( ( / ( (! d x w dx ( d / / ( / (, ( w x dx (! O diffeetiatig P times / / ( / (, ( w x (! / / / (, ( w x x (! This comletes the oof of Theoem 2. Coollay 2. Whe equatio (9 educe to Riema-Liouville factioal itegal /, / /, /,,,, D t E ( wt ( x x E ( w x (2

7 Factioal Calculus of Coollay 2.2 Whe q, equatio (9 educes i the followig fom /, / q / /, q /,,,, D t E ( wt ( x x E ( w x Coollay 2.3 Whe equatio (9 educes to (2 /, q /, q,, D t E ( wt ( x x E ( w x Coollay 2.4 Fo q ad equatio (9 educes i the followig fom D t E, ( wt ( x x E, ( w x Acowledgemet This wo is suoted by Post-Doctoal Fellowshi of the Natioal Boad of Highe Mathematics (NBHM, Deatmet of Atomic Eegy, Idia to autho (Jiteda Daiya. Refeeces [] Diaz, R., Paigua, E. (27. O hyegeometic fuctios ad Pochhamme -symbol Divulgacioes Mathematicas 5 (2, [2] Doego, G. A. Ceutti, R. A. (22. The -Mittag-Leffle fuctio, It. J. Cotem Math. Sci. 7 (5, [3] Haubold, H.J., Mathai, A.M. ad Saxea, R.K. (2. Mittag-Leffle fuctio & thei alicatios, J. Ae. Math., -5 (Aticle ID [4] Kilbas, A. A., Saigo, M. ad Saxea, R. K. (22. Solutio of voltea itego diffeetial equatios with geealized Mittag-Leffle fuctio i the eals, J. itegal Equatios Ale. 4 (22, [5] Kilbas, A. A., Saigo, M. ad Saxea, R. K. (24. Geealized Mittag-Leffle fuctio ad geealized factioal calculus oeatos, ITSF 5, [6] Mathai, A. M., Saxea, R. K. ad Haubold, H.J. (2. The H-fuctio, Theoy ad Alicatios, Sige, New Yo. [7] Mubee, S. ad Habibullah, G. M. (22. -factioal itegal ad alicatio, It. J. Cotem. Math. Sci., 7(2, [8] Romeo, L., Ceutti, R. (22. Factioal Fouie Tasfom ad Secial -Fuctio, Ite. J. Cotem. Math. Sci., Vol. 7, No. 4, (22 (23

8 96 Jiteda Daiya ad Jeta Ram [9] Romeo, L.G., Luque, L. L., Doego, G. A. ad Ceutti, R. A. (23. O the -Riema-Liouville factioal Deivative, It. J. Cotem. Math. Sci., 7(8, 4-5. [] Saxea, R. K. ad Saigo, M. (25. Cetai Poeties of factioal calculus oeato associated with geealized Mittag-Leffle fuctio, Fac. Calc. Al. Aal. 8, [] Saxea, R. K., Kalla, S. L. ad Kiyaova, V. S. (23. Relatios coectig multi-idex Mittag-Leffle fuctio ad Riema-Liouville factioal calculus, Algeba gous ad Geometies 2, [2] Saxea, R.K. (22A. Cetai oeties of geealized Mittag-Leffle fuctio, Cofeece of the Society of Secial fuctios & thei alicatios, 3 d aual Cofeece, 78-8, Cheai, Idia. [3] Saxea, R.K., Daiya, J. ad Sigh, A. (24. Itegal Tasfoms of the - γ, Mittag-Leffle fuctio E,α,β z, Le-Mathematiche, Vol. LXIX Fasc. II, 7 6. [4] Saxea, R.K., Pogáy, T.K., Ram, J. ad Daiya, J. (2. Diichlet aveages of geealized multi-idex Mittag-Lefle fuctios, Ameia joual of mathematics, Vol. 3 (4, [5] Shula, K. ad Pajaati, J. C. (27. O the geealizatio of Mittag-Leffle fuctio ad its oeties, Joual of Mathematical Aalysis ad Alicatios, 336, [6] Sivastava, H. M. ad Tomovsi, Z. (29. Factioal calculus with a itegal oeato cotaiig a geealized Mittag-Leffle fuctio i the eel, Al. Math. Comut., 2, 98-2.

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