UNIFIED FRACTIONAL INTEGRAL FORMULAE FOR THE GENERALIZED MITTAG-LEFFLER FUNCTIONS

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1 Joural o Sciece ad Ars Year 14 No OGNAL PAPE UNFED FACTONAL NTEGAL FOMULAE FO THE GENEALZED MTTAG-LEFFLE FUNCTONS DAYA LAL SUTHA 1 SUNL DUTT PUOHT 2 Mauscri received: ; Acceed aer: ; Published olie: Absrac The racioal calculus oeraors has gaied oiceable imorace ad oulariy due o is esablished alicaios i describig/modelig ad solvig various iegral euaios ordiary diereial euaios ad arial diereial euaios i may ields o sciece ad egieerig The Miag-Leler ucios are imora secial ucios ha rovides soluios o umber o roblems ormulaed i erms o racioal order diereial iegral ad dierece euaios; hereore i has recely become a subec o ieres or may auhors i he ield o racioal calculus ad is alicaios The aim o his aer is o evaluae wo uiied racioal iegrals ivolvig he roduc o geeralied Miag-Leler ucio ad Aell ucio F 3 These iegrals are urher alied i rovig wo heorems o Marichev-Saigo-Maeda racioal iegral oeraors The resuls are eressed i erms o geeralied Wrigh ucio ad hyergeomeric ucios F Furher we oi ou also heir relevace Keywords: Marichev-Saigo-Maeda racioal iegral oeraors geeralied Miag- Leler ucio geeralied Wrigh ucio geeralied hyergeomeric series 1 NTODUCTON The racioal calculus oeraors ivolvig various secial ucios have oud sigiica imorace ad alicaios i modelig o releva sysems i various ields sciece ad egieerig such as urbulece ad luid dyamics sochasic dyamical sysem lasma hysics ad corolled hermouclear usio oliear corol heory image rocessig oliear biological sysems asrohysics ad i uaum mechaics Thereore a remarkably large umber o auhors have sudied i deh he roeries alicaios ad diere eesios o various oeraors o racioal calculus For deailed accou o racioal calculus oeraors alog wih heir roeries ad alicaios oe may reer o he research moograhs by Miller ad oss [4] Samko e al [14] ad Kiryakova [2] 1 Poorima Uiversiy School o Basic ad Alied Sciece Jaiur dia dlsuhar@yahoocoi 2 MP Uiversiy Agri Tech College o Techology ad Egieerig Dearme o Basic-Scieces Mahemaics Udaiur dia suil_a_urohi@yahoocom SSN: Mahemaics Secio

2 118 Uiied racioal iegral ormulae or Daya Lal Suha Suil Du Purohi 1903 Gosa Miag-Leler [5] iroduced he ucio E deied by: 1 E 1 C 0 ; e > 0 11 A urher wo-ide geeraliaio o his ucio was give by Wima [17] as: 1 E C 0 12 where e > 0 ad e > 0 By meas o he series rereseaio a geeraliaio o Miag-Leler ucio 12 is iroduced by Prabhakar [7] as: E 13! 0 1 where C e > 0 Furher i is a eire ucio o order [e ] [7] Sice he Miag-Leler ucio rovides soluios o cerai roblems ormulaed i erms o racioal order diereial iegral ad dierece euaios hereore a umber o useul geeraliaio o he his ucio has bee iroduced ad sudied may auhors ecely Salim ad Fara [13] has iroduced ad sudied a ew geeraliaio o he Miag- Leler ucio by meas o he ower series: E 14 0 where C; > 0 such ha The geeralied Wrigh hyergeomeric ucio ψ or C comle a b C ad 0; i 12 ; 12 is deied as below: i i i a i i 1 i1 ψ ψ b 1 k 0 b k k! k ai ik 15 1 Wrigh [18] iroduced he geeralied Wrigh ucio 15 ad roved several heorems o he asymoic easio o ψ [18-20] or all values o he argume uder he codiio: 1 > 1 i i1 wwwosaro Mahemaics Secio

3 Uiied racioal iegral ormulae or Daya Lal Suha Suil Du Purohi 119 The geeralied hyergeomeric ucio or comle a b C b 0 1 i 12 ; 12 is give by he ower series [1]: i ad F a a ; b b ; 1 1 r a1 r a r 16 b b r! r 0 1 r r where or covergece we have < 1 i 1 ad or ay i The ucio 16 is a secial case o he geeralied Wrigh ucio 15 or 1 1 1: b a 1 1 i 1 F a1 a; b1 b; ψ b ai i1 A useul geeraliaio o he hyergeomeric racioal iegrals icludig he Saigo oeraors [10-11] has bee iroduced by Marichev [3] see deails i Samko e al [14] ad laer eeded ad sudied by Saigo ad Maeda [12] i erm o ay comle order wih Aell ucio F 3 i he kerel as ollows: Le C ad >0 he he geeralied racioal calculus oeraors he Marichev-Saigo-Maeda oeraors ivolvig he Aell ucio or Hor's F 3 -ucio are deied by he ollowig euaios: 1 0 F 0 3 ; ;1 1 d > F3 ; ;1 1 d >0 19 For he deiiio o he Aell ucio F 3 he ieresed reader may reer o he moograh by Srivasava ad Karlso [16] see [1 6] Followig Saigo e al [10 15] he image ormulas or a ower ucio uder oeraors 18 ad 19 are give by: where > ma{ 0 } ad > wher < 1 mi{ } ad >0 The symbol occurrig i 110 ad 111 is give by: a b c d e a b c d e SSN: Mahemaics Secio

4 120 Uiied racioal iegral ormulae or Daya Lal Suha Suil Du Purohi The comuaios o racioal iegrals ad racioal derivaives o secial ucios o oe ad more variables are imora rom he oi o view o he useuless o hese resuls i he evaluaio o geeralied iegrals ad he soluio o diereial ad iegral euaios or eamle see [8]-[9] Moivaed by hese aveues o alicaios here we esablish wo image ormulas or he geeralied Miag-Leler ucio 14 ivolvig le ad righ sided oeraors o Saigo-Meada racioal iegral oeraors [12] i erm o he geeralied Wrigh ucio [18] 2 MAN ESULTS his secio we esablish image ormulas or he geeralied Miag-Leler ucio ivolvig le ad righ sided oeraors o Saigo-Meada racioal iegral oeraors 18 ad 19 i erm o he geeralied Wrigh ucio These ormulas are give by he ollowig heorems: Theorem 21 Le C ad > 0 > 0 be such ha >0 > 1 > ma 0 he here hold he ormula: [ ] 1 E 0 c [ ] ψ 4 c 21 Proo: O usig 14 ad wriig he ucio i he series orm he le had side o 21 leads o 1 c 1 E [ c ] Now uo usig he image ormula 110 which is valid uder he codiios saed wih Theorem 21 we ge 1 E [ c ] c! 23 erreig he righ-had side o he above euaio i view o he deiiio 15 we arrive a he resul 21 wwwosaro Mahemaics Secio

5 Uiied racioal iegral ormulae or Daya Lal Suha Suil Du Purohi SSN: Mahemaics Secio 121 Theorem 22 Le C ad 0 > such ha [ ] < 1 1 > > 0 mi he he ollowig ormula holds rue: ] [ 0 c E ψ c 24 Proo: By usig 14 he le had side o 24 ca be wrie as: ] [ c c E 25 which o usig he image ormula 111 arrive a ] [ 0 c E 0! 1 c r 26 erreig he righ-had side o he above euaio i view o he deiiio 15 we arrive a he resul 24 3 SPECAL CASES his secio we cosider some secial cases o he mai resuls derived i he recedig secio we se 0 i he oeraors 18 ad 19 he we have he ollowig kow ideiies: where he hyergeomeric oeraors aeared i he righ had side are due o Saigo[10] deied as: / ;1 ; d F 33

6 122 Uiied racioal iegral ormulae or Daya Lal Suha Suil Du Purohi 1 1 2F ; ;1 / 0 1 d 34 Thereore i we se 0 ad relace by i 21 ad 22 we ge he ollowig resuls ivolvig he le ad righ had sided Saigo ye iegral oeraors: Corollary 31 Le C ad > 0 >0 >0 he here hold he ormula: 1 E [ c ] ψ 3 c 35 Corollary 32 Le C ad > 0 > ma[ ] he he ollowig ormula hold: E [ c ] ψ 4 c 36 Furher i we ollow resuls o Corollaries 31 ad 32 whe we arrive a he ollowig resuls ivolvig le ad righ sided iema-liouville racioal iegraio oeraor Corollary 33 C ad > 0 >0 >0 he we have 1 E [ c ] ψ 2 c 37 emark 1 we se 1 i euaio 37 we ge he kow resul give by Saea ad Saigo [15] Corollary 34 Le C ad > 0 > ma[ ] he W E 0 c [ ] ψ c 38 emark 2 we se 1 i euaio 38 we ge he kow resul give by Saea ad Saigo [15] wwwosaro Mahemaics Secio

7 Uiied racioal iegral ormulae or Daya Lal Suha Suil Du Purohi 123 Fially i we ollow Corollaries 31 ad 32 i resecive case 0 he we arrive a he ollowig corollary cocerig le ad righ sided Erdélyi-Kober racioal iegraio oeraors Corollary 35 Le C ad > 0 >0 >0 he here hold he ormula: 1 E E [ c ] ψ 3 c 39 Corollary 36 Le C ad > 0 > ma[ ] he here hold he ormula 1 1 K0 E [ c ] 4ψ 4 c 310 EFEENCES [1] Erdélyi A Magus W Oberheiger F Tricomi FG Higher Trascedeal Fucios Vol1 McGraw-Hill New York 1953 [2] Kiryakova V Geeralied Fracioal Calculus ad Alicaios Vol 301 Logma Scieiic & Techical Esse UK 1994 [3] Marichev O v AN BSS Ser Fi-Ma Nauk [4] Miller KS oss B A roducio o he Fracioal Calculus ad Fracioal Diereial Euaios Wiley-er-Sciece Joh Wiley & Sos New York USA 1993 [5] Miag-Leer GM C Acad Sci Paris [6] Prudikov AP Brychkov YuA Marichev O egrals ad Series Secial Fucios Vol 1-5 Gordo & Breach New York 1992 [7] Prabhakar T Yokohama Mah J [8] Purohi SD Kalla SL Suhar DL SCENTA Series A: Mahemaical Scieces [9] Purohi SD Suhar DL Kalla SL Le Maemaiche [10] Saigo M Mah e Kyushu Uiv [11] Saigo MA Mah Jaoica [12] Saigo M Maeda N More geeraliaio o racioal calculus : Trasorm Meods ad Secial Fucios Soia [13] Salim TO Fara AW Joural o Fracioal Calculus ad Alicaios [14] Samko S Kilbas A Marichev O Fracioal egrals ad Derivaives Theory ad Alicaios Gordo & Breach Sci Publ New York 1993 [15] Saea K Saigo M J Frac Calc [16] Srivasava HM Karlso PW Mulile Gaussio Hyergeomeric Series Ellis Horwood Limied New York 1985 SSN: Mahemaics Secio

8 124 Uiied racioal iegral ormulae or Daya Lal Suha Suil Du Purohi [17] Wima A Aca Mah [18] Wrigh EM J Lodo Mah Soc [19] Wrigh EM Philos Tras oy Soc Lodo A [20] Wrigh EM Proc Lodo Mah Soc wwwosaro Mahemaics Secio

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