Unified Mittag-Leffler Function and Extended Riemann-Liouville Fractional Derivative Operator

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1 Iteatioal Joual of Mathematic Reeach. ISSN Volume 9, Numbe 2 (2017), pp Iteatioal Reeach Publicatio Houe Uified Mittag-Leffle Fuctio ad Exteded Riema-Liouville Factioal Deivative Opeato S. C. Shama ad Meu Devi Depatmet of Mathematic, Uiveity of Rajatha, Jaipu , Rajatha, Idia. Abtact I thi pape we tudy cetai popetie of exteded Riema-Liouville factioal deivative opeato aociated with uified Mittag-Leffle fuctio i the fom of theoem. Futhe we have eumeated ome eult of Geealied Mittag-Leffle fuctio ad Riema-Liouville factioal deivative opeato a pecial cae. 1. Itoductio ad Pelimiaie Uified Mittag-Leffle fuctio The Swedih mathematicia Gota Mittag-Leffle [1] itoduced the fuctio E () i 1903 i the fom E ( ),, C; Re( ) 0 ( 1) (1) 0 Due to ivolvemet of Mittag Leffle fuctio i poblem of Phyic, Biology, Egieeig, Applied ciece ad i the olutio of factioal ode diffeetial o itegal equatio, umeou eeache defied ad tudied vaiou geealiatio E, by Wimo [2], E,, q,, by Shukla ad Pajapati [4], E,,, q of Mittag Leffle type fuctio a E E,, p by Salim ad Faa [6]. by Pabhaka [3], by Sivatava ad Tomovki [5] ad

2 136 S. C. Shama ad Meu Devi Recetly, Shukla ad Pajapati [7] itoduced a ew geealiatio of Mittag-Leffle fuctio amed Uified Mittag Leffle fuctio i the fom p1 ( ), ( c) E,,,,, p( c;, ) (2) 0 ( p 1) ( ) ( ) p whee,,,,, C; Re(,,,, ) 0;,, p, c 0 ( q) ad ( ) q, i the geealied Pochhamme ymbol. ( ) Due to igificat applicatio i vaiou divee field of ciece ad egieeig, factioal calculu ha gaied coideable impotace duig pat fou decade [8-14]. Hece i ecet yea, cetai exteded factioal deivative opeato have bee itoduced ad tudied [15, 16]. Recetly, Agawal, Choi ad Pai itoduced a ew exteio of Riema-Liouville factioal deivative opeato [19]. I thi pape we will tudy cetai popetie of thi exteded Riema-Liouville factioal deivative opeato with uified Mittag-Leffle fuctio. Befoe Defiig exteded Riema-Liouville factioal deivative opeato, we ae givig ome defiitio:, ;, Defiitio 1.1 The exteded Beta fuctio B ( x, y) i give a [17]: 1, ;, x1 y1 p Bp ( x, y) t (1 t) 1F1 ; ; dt k t (1 t) 0 whee 0, 0, R( p) 0,mi R( ), R( ) 0, R( x) R( ), R( y) R( ) Defiitio 1.2 The exteded Gau hypegeometic fuctio i give a [19]: p (3), ;, B (, ), ;, p b c b Fp ( a, b; c; ) ( a) (4) B( b, c b)! 0 whee 1,mi R( ), R( ), R( ), R( ) 0, R( c) R( b) 0, R( p) 0 Defiitio 1.3 A exteio of the exteded Gau hypegeometic i give a [19]:, ;, F p fuctio F p;,, ;, ( a) ( ) (, ) b Bp b c b m ( a, b; c; ; m) (5) ( c) B( b, c b m)! 0 whee 1, p 0, R( ) 0, R( ) 0, m R( b) R( c) Defiitio 1.4 A exteio of the exteded Appell hypegeometic fuctio F 1 i give a [19]:, ;, k ( a) ( ) ( ) (, ) k b c Bp a k d a m k x y F1, p;, ( a, b, c; d; x, y; m) (6) ( d) B( a k, d a m)! k! k, 0 k

3 Uified Mittag-Leffle Fuctio ad Exteded Riema-Liouville Factioal 137 whee x 1, y 1, p 0, R( ) 0, R( ) 0, m R( a) R( d) Defiitio 1.5 A exteio of the Appell hypegeometic fuctio F 2 i give a [19]: F 2, p;, ( a, b, c; d, e; x, y; m), ;,, ;, k ( a) ( ) ( ) (, ) (, ) k b c Bp b d b m B k p c k e c m x ( d) ( e) B( b, d b m) B( c k, e c m)! k! k, 0 k whee x y 1, p 0, R( ) 0, R( ) 0, m R( b) R( d), m R( c) R( e) Defiitio 1.6 A exteio of the Lauicella hypegeometic fuctio [19]: F 3 Dp, ;, ( a, b, c, d; e; x, y, ; m), k, 0, ;, k ( a) ( ) ( ) ( ) (, ) k b c k d B p a k e a m x y ( e) B( a k, e a m)! k!! k whee x 1, y 1, 1, p 0, R( ) 0, R( ) 0, m R( a) R( e) (7) 3 F D i give a Defiitio 1.7 The claical Riema-Liouville factioal deivative opeato of ode i give a [20]: 1 1 ( ) ( ) ( ), ( ) 0 D f t f t dt R ( ) 0 m d m ad D f ( ) D f ( ), R( ) 0, m 1 R( ) m (10) m d (8) (9) Exteded Riema-Liouville factioal deivative opeato The exteded Riema-Liouville factioal deivative opeato of ode i defied a [19]:, p;, 1 1 p ( ) ( ) ( ) 1 1 ; ; ( ) t ( t) (11) 0 D f t f t F dt whee R( ) 0, R( p) 0, R( ) 0, R( ) 0 Fo R( ) 0 m, p;, d m, p;, D f ( ) D f ( ) m d m 1 R( ) m m N, R( p) 0, R( ) 0, R( ) 0 whee (12)

4 138 S. C. Shama ad Meu Devi 2. MAIN RESULTS ad R( ) R( ), we have [19]: Lemma 1. Fo R( ) 0, m 1 R( ) mm N, ;, ( 1, ), p;, ( 1) Bp m D ( 1) B( 1, m ) (13) Theoem 1. Fo R( ) 0, m 1 R( ) mm N ad R( ) R' 1 D E c ;,, p;, ' 1, ' ', ',, ',, ' 1 0 ' p ' 1, ;, p ' p ' 1 ' ', we have: ( ) c B ' p ' 1 ', m B ' p ' 1 ', m Poof. Uig defiitio of uified Mittag-Leffle fuctio, we get: ' p ' 1 ( ), ;, ' 1, ', ;, ' 1 c p p D E ', ',, ',, c ;, D 0 '( p ' 1) ' ( ) ' ( ) Afte itechagig the ole of ummatio ad exteded Riema-Liouville factioal deivative opeato, we get: D E c ;,, p;, ' 1, ' ', ',, ',, (14) 0 ( ) c p ' 1, p;, p p ' '( ' 1) ' ( ) ( ) uig Lemma 1 ad afte ome implificatio, we get: D E c ;,, p;, ' 1, ' ' 1 ', ',, ',, 0 D ' ' 1 ' 1 p ' 1, ;, c Bp p m ' p ' 1 ( ) ' ' 1 ', '( p ' 1) ' ( ) ' ' 1 ', ' ( ) B p m which i equied eult.

5 Uified Mittag-Leffle Fuctio ad Exteded Riema-Liouville Factioal 139 Remak 1.1 If we take p 0 i equatio (14), we obtai thi eult fo uified Mittag-Leffle fuctio ad claical Riema-Liouville factioal deivative opeato. Remak 1.2 If we take c 1, 0, q i equatio (14), we obtai thi eult fo geealied Mittag-Leffle fuctio. ad R( ) R( ' 1), we have: Theoem 2. Fo R( ) 0, m 1 R( ) mm N 1 ', ',, ',, ;, D E c, p;, ' 1, ' ' 1 0 ' ( ) c p ' 1 ' ' 1 ' ' p (15) Fp;,, ' p ' 1 '; ' p ' 1 ' ; ; m Poof. Uig defiitio of uified Mittag-Leffle fuctio, we get: 1 ', ',, ',, ;, D E c, p;, ' 1, ', p;, ' 1 D 1 0 uig biomial eie expaio, we get: 1 ', ',, ',, ;, D E c, p;, ' 1, ' ( ) c ' p ' 1 '( p ' 1) ' ( ) ' ( ) ' p ' 1 ( ), ;, ' 1 ( ) c p l l D l0 l! 0 '( p ' 1) ' ( ) ' ( ) Afte itechagig the ole of ummatio ad exteded Riema-Liouville factioal deivative opeato, we get: 1 ', ',, ',, ;, D E c, p;, ' 1, '

6 140 S. C. Shama ad Meu Devi p ' 1 ( ) c ( ) l, p;, '( 1) ' l1 D l! ' 0 '( ' 1) ' p ( ) ( ) l0 uig Lemma 1 ad afte ome implificatio, we get: 1 ', ',, ',, ;, D E c, p;, ' 1, ' 1 '( 1) ' 1 ( ) c ( ) '( p ' 1) ' l l 0 '( ' 1) ' ( ) 0! '( ' 1) ' p ' ( ) l p l l (16), ;, Bp ' p ' 1 ' l, m B ' p ' 1 ' l, m uig exteio of the exteded Gau hypegeometic fuctio give by equatio (5), we get: 1 ', ',, ',, ;, D E c, p;, ' 1, ' ' 1 0 p;, ( ) ' p ' 1 p ' c '( ' 1) ' ( ) ( ) F, '( p ' 1) '; '( p ' 1) ' ; ; m which i equied eult. Remak 2.1 If we take p 0 i equatio (14), we obtai thi eult fo uified Mittag-Leffle fuctio ad claical Riema-Liouville factioal deivative opeato. Remak 2.2 If we take c 1, 0, q i equatio (14), we obtai thi eult fo geealied Mittag-Leffle fuctio. Theoem 3. Fo R( ) 0, m 1 R( ) mm N l ad R( ) R( ' 1), we have:

7 Uified Mittag-Leffle Fuctio ad Exteded Riema-Liouville Factioal ', ',, ',, ;, D a b E c, p;, ' 1, ' ' 1 0 ' ( ) c p ' 1 ' ' 1 ' ' p F1, p;, ' 1 ',, ; ' 1 ' ; a, b; m (17) Poof. Uig defiitio of uified Mittag-Leffle fuctio, we get: 1 1 ', ',, ',, ;, D a b E c, p;, ' 1, ', p;, ' 1 D a b ( ) uig biomial eie expaio, we get: 1 1 ', ',, ',, ;, D a b E c, p;, ' 1, ' 1 l2 1 2 c ' p ' 1 '( p ' 1) ' ( ) ' ( ) ' p ' 1 ( ), ;, ' 1 l l l ( ) c p D a b l1, l20 l1! l2! 0 '( p ' 1) ' ( ) ' ( ) Afte itechagig the ole of ummatio ad exteded Riema-Liouville factioal deivative opeato, we get: 1 1 ', ',, ',, ;, D a b E c, p;, ' 1, ' p ' 1 ( ) ( ) c l 1 l l l 2 p l1 l2 0 '( p ' 1) ' ( ) 1, 2 0 ' ( ) l l l ' 1! l2! p 1 2, p;, '( ' 1) ' 1 a b D uig Lemma 1 ad afte ome implificatio, we get: 1 1 ', ',, ',, ;, D a b E c, p;, ' 1, '

8 142 S. C. Shama ad Meu Devi ( ) c 1 '( 1) ' 1 '( p ' 1) ' ( ) l l1l2 1 l2 '( p ' 1) ' 0 '( p ' 1) ' ( ) 1, 2 0 ' ( ) l l l 1 l 2 l l, ;, B p ' p ' 1 ' l1 l2, m a b B ' p ' 1 ' l1 l2, m l1! l2! 1 2 uig exteio of the exteded Appell hypegeometic fuctio give by equatio (6), we get: 1 1 ', ',, ',, ;, D a b E c, p;, ' 1, ' ' 1 0 which i equied eult. ' ( ) c p ' 1 ' ' 1 ' ' p F1, p;, ' 1 ',, ; ' 1 ' ; a, b; m Remak 3.1 If we take p 0 i equatio (14), we obtai thi eult fo uified Mittag-Leffle fuctio ad claical Riema-Liouville factioal deivative opeato. Remak 3.2 If we take c 1, 0, q i equatio (14), we obtai thi eult fo geealied Mittag-Leffle fuctio. ad R( ) R( ' 1), we have: Theoem 4. Fo R( ) 0, m 1 R( ) mm N ', ',, ',, ;, D a b d E c, p;, ' 1, ' ' 1 0 ' ( ) c p ' 1 ' ' 1 ' ' p 3 FDp, ;, ' 1 ',,, ; ' 1 ' ; a, b, d; m (18)

9 Uified Mittag-Leffle Fuctio ad Exteded Riema-Liouville Factioal 143 Poof. Uig defiitio of uified Mittag-Leffle fuctio, we get: ', ',, ',, ;, D a b d E c, p;, ' 1, ', p;, ' 1 D a b d ( ) c ' p ' 1 '( p ' 1) ' ( ) ' ( ) uig biomial eie expaio, we get: ', ',, ',, ;, D a b d E c, p;, ' 1, ' ( ) D a b d, p;, ' 1 l 1 l2 l l 3 1 l2 l3 l1, l2, l3 0 l1! l2! l3 0 ' p ' 1 ( ) c ' '( p ' 1) ' ( ) ( ) Afte itechagig the ole of ummatio ad exteded Riema-Liouville factioal deivative opeato, we get: ', ',, ',, ;, D a b d E c, p;, ' 1, ' p ' 1 ( ) ( ) c l 1 l2 l l 3 1 l2 l3 a b d 0 '( p ' 1) ' ( ) 1, 2, 3 0 ' ( ) l l l l ' 1! l2! l p 3 D, p;, '( 1) ' l1 l2 l3 1

10 144 S. C. Shama ad Meu Devi uig Lemma 1 ad afte ome implificatio, we get: ', ',, ',, ;, D a b d E c, p;, ' 1, ' 0 ( ) c 1 '( 1) ' 1 p ' '( ' 1) ' ( ) ( ) '( p ' 1) ' ( ) l l1 l2 l3 1 l2 l3 '( p ' 1) ' l1, l2, l 3 0 l1 l2 l3 l l l, ;, B p ' p ' 1 ' l1 l2 l3, m a b d B ' p ' 1 ' l1 l2 l3, m l1! l2! l3! uig exteio of the exteded Lauicella hypegeometic fuctio give by equatio (8), we get: ', ',, ',, ;, D a b d E c, p;, ' 1, ' ' 1 0 ' ( ) c p ' 1 ' ' 1 ' ' p 3 FDp, ;, ' 1 ',,, ; ' 1 ' ; a, b, d; m which i equied eult. Remak 4.1 If we take p 0 i equatio (14), we obtai thi eult fo uified Mittag-Leffle fuctio ad claical Riema-Liouville factioal deivative opeato. Remak 4.2 If we take c 1, 0, q i equatio (14), we obtai thi eult fo geealied Mittag-Leffle fuctio.

11 Uified Mittag-Leffle Fuctio ad Exteded Riema-Liouville Factioal 145 ad R( ) R( ' 1), we have: Theoem 5. Fo R( ) 0, m 1 R( ) mm N, p ;, ' 1 x 1, D ' Fp;,, ; ; ; me ', ',, ',, c ;, 1 ' 1 0 ' ( ) c p ' 1 ' ' 1 ' ' p F2, p;,,, ' 1 ';, ' 1 ' ; x, ; m (19) Poof. Uig defiitio of exteio of the exteded Gau hypegeometic fuctio, we get:, p ;, ' 1 x 1, D ' Fp;,, ; ; ; me ', ',, ',, c ;, 1, p;, ' 1, ' D E c 1 ', ',, ',, ;,, ;, l 1 l1 p 1, l1 B l m x ( ) l! B l, m 1 l1 0 l1 1 1 Afte itechagig the ole of ummatio ad exteded Riema-Liouville factioal deivative opeato, we get:, p ;, ' 1 x 1, D ' Fp;,, ; ; ; me ', ',, ',, c ;, 1 l1, ;,, 1 1 ( ) l! B l, m x B l m D E c l l p 1, p;, ' 1, ' l1 0 l1 1 1 l1 1 ', ',, ',, ;, uig eult of Theoem 2 give by equatio (16) ad afte ome implificatio, we get:, p ;, ' 1 x 1, D ' Fp;,, ; ; ; me ', ',, ',, c ;, 1 l l p 2 '( p 1) ' ' p ' 1 ( ) ( ) '( 1) ' ' 1 c 1 l1 l2 0 '( ' 1) ' p ( ) 1, 2 0 ' ( ) l l l 1 l 2

12 146 S. C. Shama ad Meu Devi, ;,, ;, l1 l B ' ' 1 ' 2 1, 1, p l m Bp p l m x B l1, m B ' p ' 1 ' l1, m l1! l2! uig exteio of the exteded Appell hypegeometic fuctio give by equatio (7), we get:, p ;, ' 1 x 1, D ' Fp;,, ; ; ; me ', ',, ',, c ;, 1 ' 1 0 ' ( ) c p p ' 1 ' ' 1 ' ' F2, p;,,, ' 1 ';, ' 1 ' ; x, ; m which i equied eult. Remak 5.1 If we take p 0 i equatio (14), we obtai thi eult fo uified Mittag-Leffle fuctio ad claical Riema-Liouville factioal deivative opeato. Remak 5.2 If we take c 1, 0, q i equatio (14), we obtai thi eult fo geealied Mittag-Leffle fuctio. REFERENCES [1] Mittag-Leffle, G. M., 1903, Su la ouvelle foctio Eα(x), C. R. Acad. Sci. Pai, 137, pp [2] Wima, A., 1905, Ube de fudametal at i de theoie de fuktioe Eα(x), Acta Math, 29, pp [3] Pabhaka, T. R., 1971, A igula itegal equatio with a geealied Mittag-Leffle fuctio i the keel, Yokohama Math. J., 19, pp [4] Shukla, A. K., ad Pajapati, J. C., 2007, O a geealiatio of Mittag- Leffle fuctio ad it popetie, Joual of Mathematical Aalyi ad Applicatio, 336(2), pp

13 Uified Mittag-Leffle Fuctio ad Exteded Riema-Liouville Factioal 147 [5] Sivatava, H. M., ad Tomovki, Z., 2009, Factioal calculu with a itegal opeato cotaiig a Geealied Mittag-Leffle fuctio i the keel, Applied Mathematic ad Computatio, 211(1), pp [6] Salim, T. O., ad Faaj, A. W., 2012, A geealiatio of Mittag-Leffle fuctio ad itegal opeato aociated with factioal calculu. Joual of Factioal Calculu ad Applicatio, 3(5), pp [7] Pajapati, J. C., Dave, B. I., ad Nathwai, B. V., 2013, O a uificatio of geealied Mittag-Leffle fuctio ad family of Beel fuctio, Adv. Pue Math., 3(1), pp [8] Almeida, R., ad Toe, D. F. M., 2011, Neceay ad ufficiet coditio fo the factioal calculu of vaiatio with Caputo deivative, Noliea Sci. Nume. Simul., 16, pp [9] Kilba, A. A., Sivatava, H. M., ad Tujillo, J. J., 2006, Theoy ad applicatio of factioal diffeetial equatio, Noth Hollad Mathematic Studie, vol. 204, Elevie Sciece B. V., Amtedam. [10] Li, C., Che, A., ad Ye, J., 2011, Numeical appoache to factioal calculu ad factioal odiay diffeetial equatio, J. Comput. Phyic, 230, pp [11] Machado, J. L., Kiyakova, V., ad Maiadi, F., 2011, Recet hitoy of factioal calculu, Commu. Noliea Sci. Nume. Simulat., 16, pp [12] Magi, R. L., 2010, Factioal calculu model of complex dyamic i biological tiue, Comput. Math. Appl., 59, pp [13] Mathai, A. M., Saxea, R. K., ad Houbold, H. J., 2010, The H-fuctio. Theoy ad applicatio, Spige, New Yok. [14] Zhao, J., 2015, Poitive olutio fo a cla q-factioal bouday value poblem with p - Laplacia, J. Noliea Sci. Appl., 8, pp [15] Luo, M. J., Milovaovic, G. V., ad Agawal, P., 2014, Some eult o the exteded beta ad exteded hypegeometic fuctio, Appl. Math. Comput., 248, pp [16] Oala, M. A., ad Oegi, E., 2010, Some geeatig elatio fo exteded hypegeometic fuctio via geealied factioal deivative opeato, Math. Comput. Model, 52, pp [17] Sivatava, H. M., Agawal, P., ad Jai, S., 2014, Geaatig fuctio fo the geealied cla hypegeometic fuctio, Appl. Math. Comput., 247, pp

14 148 S. C. Shama ad Meu Devi [18] Sivatava, H. M., ad Choi, J., 2012, Zeta ad q-zeta fuctio ad aociated eie ad itegal, Elevie ciece publihe, Amtedam, Lodo ad New Yok. [19] Agawal, P., Choi, J., ad Pai, R. B., 2015, Exteded Riema-Liouville factioal deivative opeato ad it applicatio, J. Noliea Sci. Appl., 8, [20] Samko, S. G., Kilba, A. A., ad Maichev, O. I., 1993, Factioal Itegal ad Deivative: Theoy ad Applicatio, Godo & Beach, New Yok.