A REMARKABLE SURVEY ON GENERALIZED MITTAG-LEFFLER FUNCTION AND APPLICATIONS J. C. PRAJAPATI 1 and A. K. SHUKLA 2

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Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ RMRKBL SURVY ON GNRLIZD MITTG-LFFLR FUNCTION ND PPLICTIONS J. C. PRJPTI d. K. SHUKL Dee of Mheicl Sciece Fculy of lied Sciece Cho Uiveiy of Sciece d Techology Chg d-388 4 Idi. Dee of lied Mheic d Huiie S.V.Niol Iiue of Techology Su-395 7 Idi -il: jyoid8@ediffil.co jyoidji.h@chu.c.i jyhul@ediffil.co bc. The icil i of he e i o ioduce he viou geeliio of Shul-Pji fucio d olyoil. Fo hee ew fucio d olyoil hei viou oeie icludig uul diffeeiio d iegio Iegl fo Geelied hyegeoeic eie fo Melli Be iegl eeeio Recuece elio Iegl eeeio Decooiio Fciol clculu oeo oeie geeig elio bilel geeig elio d fiie uio foule of ew cl of olyoil lo eblihed. Key Wod: Mig Leffle fucio Iegl Tfo Fciol iegl d diffeeil oeo. MS clificio : 33 33C 44 3. INTRODUCTION ND PRLIMINRIS Recely Shul d Pji 7 iveiged d udied he fucio q which i defied fo q : Re > Re > Re > d q! q.. 38

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ whee q q deoe he geelied Pochhe ybol Riville96 which i icul educe o q q q q if q. I coiuio of he udy he geeliio of c be wie Re Re Re > d > which i defied fo > >!.. Thi i geeliio of he eoeil fucio e he coflue hyegeoeic fucio Φ Riville 96 he Mig Leffle fucio Mig-Leffle 93 he Wi fucio Wi 95 d he fucio defied by Pbh 97 well equio.. Goeflo e l. 998 Goeflo d Midi Kilb e l. 996 4 Sigo d Kilb 998 Sivv d Toovi 9 Toovi e l d y ohe eeche lo udied he viou oeie of Mig-Leffle fucio d i geeliio wih hei licio. The licio of Iegl Tfo dicued by Seddo 979. The fucio i eie fucio of ode Re if Re > d boluely covege i { < R R < } if Re. The uced owe eie of he fucio c be defied N N! N j j j.3 d ecil ce fo he udy of q fo :! j whee N Re > Re > d >. j j..4 39

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ uho iveiged he oeo w f w..5 f [ w ] f d > whee w Re > Re > Re > >. If he.5 educe o he eul Pbh 97. Shul d Pji 8 ioduced cl of olyoil which e coeced by Mig-Leffle fucio i coiuio of he udy of cl of olyoil uho ioduced geel cl of olyoil defied! [ ].6 { } θ { } whee e el o cole ube e co d geelied Mig-Leffle fucio defied.. The oof of ll eul eblihed i hi e e llel o Shul d Pji 7 7 B 7 C 8 8 B 9 9 B. BSIC PROPORTIS OF TH FUNCTION coequece of he defiiio. he followig eul hold: THORM.. If Re > Re > Re > d > he i icul d d.!. THORM.. If w > he fo..3 Re > Re > Re > d 4

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ 4 d d.4 w d d Re >.5 i icul w d d.6 d w w d d Φ Φ.7 THORM.3. If wih elively ie d > he d d!.8 i icul e 3.....9 3 GNRLIZD HYPRGOMTRIC FUNCTION RPRSNTTION OF q Uig. ig N d N he we hve F. 3.

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ Covegece ciei fo geelied hyegeoeic fucio F : i If he fucio F covege fo ll fiie. ii If he fucio F covege fo < d divege fo >. iii If > he fucio F divege fo. iv If he fucio F i boluely covege o he cicle if j i Re >. j i whee i -ule i -ule....... 4 MLLIN-BRNS INTGRL RPRSNTTION OF THORM 4.. Le R d fucio i eeeed by he Melli Be iegl : N. The q d i π L whee g < π he coou of iegio begiig i i d ideed o ee he ole of he iegd fo ll o he lef fo hoe fo ll N o he igh. q 4. 5 INTGRL TRNSFORMS OF d edig I hi ecio oe ueful iegl fo lie ule fo Llce fo Melli fo d Whie fo lo dicued. N 4

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ THORM 5.. ule Be fo b σ b σ b σ d ψ whee b Re σ > d >. Re > Re > Re > Re > Re > 5. THORM 5.. Llce fo. d ψ σ σ e Whee σ d <. σ σ Re > Re > Re > Re > Re > > THORM 5.3. Melli fo. 5. w d w 5.3 d > whee Re > Re > Re > Re > To obi Whie fo we ue he followig iegl whee Re ± µ. µ v µ v d e v Wλ µ λ v v >. 5.4 THORM 5.4. Whie Tfo. 43

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ ρ e W λ µ ρ 3 ψ w d q ± µ ρ λ ρ w 5.5 > ρ d >. whee Re Re > Re > Re > Re > 6 RCURRNC RLTIONS > THORM 6.. Fo y Re > Re > d Re > we ge 3 3 { } 3 6. d d d d q q [ ] d q q whee [ ] THORM 6.. Fo d N. [ ] 3 3 3. 6. 7 INTGRL RPRSNTTIONS THORM 7.. Fo y Re > Re > d Re > d eig whee d N he > d 7. THORM 7.. If Re > Re > Re > Re > d > he 44

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ λ Secil ce of Theoe 7.: Fo Re > below: e d λ. 7. fo 7. he icul ce lied d 7.3 coh d 7.4 ih d 7.5 e ef d THORM 7.3. If Re > Re > Re > d > c. 7.6 he d [ ] THORM 7.4. If Re > Re > Re > d > e 4 he d π THORM 7.5. If Re > Re > Re > d > d d. 7.7. 7.8 he [ ] [ ] THORM 7.6. If Re > Re > Re > d > he.7.9 45

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ L. 7. THORM 7.7. If Re > Re > Re > > > d > he e q! d. 7. q d. 7. q q d. 7.3 THORM 7.8. If Re µ > Re > λ d > he If he Re > Re > Re > Re > u u u du λ d. 7.4 λ. 7.5 w w d w v µ µ v µ v µ v i icul 7.6 w d w. 7.7. 46

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ w d w 7.8 d Φ w d Φ w. 7.9 8 DCOMPOSITION OF MITTG-LFFLR FUNCTION THORM 8.. Iegl Reeeio of he fucio whee Re > Re > THORM 8.. If d >. Re > Re > u u du 8. he d > d d!! 8. THORM 8.3. Re 8.3 whee Re > Re >. d > Re of Theoe 8.3: I i ey o veify h N N. 8.4 47

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ THORM 8.4. If Re > Re > d > he. N T 8.5 whee N N N N T Re. N N N! N 8.6 9 FRCTIONL INTGRL ND DIFFRNTIL OPRTORS SSOCITDWITH TH FUNCTION The followig well-ow fc e eed fo udyig oeie of he Rie- Liouville fciol iegl d diffeeil oeo ocied wih he fucio q d lo he oeie of oeo w f. L b Sce of Lebegue euble el o cole vlued fucio Kilb e l.4: L b Coi of Lebegue euble el o cole vlued fucio f o [ b] i.e. L b b f : f f d <. 9. Kilb e l. 4 udied he evel oeie of fciol iegl d diffeeil iegl oeo. Coflue hyegeoeic fucio Riville 96: Thi i lo ow he Pochhe Be coflue hyegeoeic fucio defied Φ b F b 9. b! whee b o egive iege i covege fo ll fiie. Gu ulilicio heoe Riville 96: If i oiive iege d he 48

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ π. 9.3 Rie-Liouville fciol iegl of ode µ Kh d buhh GS 3 Le f L b µ Re µ > he µ I f f I µ d µ µ f > 9.4 i clled R-L lef-ided fciol iegl of ode µ. Le f L b µ Re µ > he µ I b f f I b µ d µ µ f b < b 9.5 i clled R-L igh-ided fciol iegl of ode µ. THORM 9.. Le R [ d le w Re Re Re Re µ > > fo > he d µ µ [ { w } I µ [ { w } D [ w µ THORM 9.. Le µ λ d > he ] 9.6 [ w ]. µ µ I [ λ λ ] λ λ µ µ 9.7 9.8 µ i icul µ λ I e µ µ λ. 9.9 THORM 9.3. Le R [ d le w Re Re Re Re µ > > fo > he 49

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ q w µ µ q µ w µ. 9. THORM 9.4. Le R [ d le w Re Re Re q Re µ > > fob > he he oeo w i bouded o L b d q w f B f 9. Whee Re B b q w b [Re Re ]! Re 9. he elio I f µ q q w µ w 9.3 f hold fo y uble fucio f L b. We c ee he fucio N : π!. 9.4 O ubiuig i 9.4 d he he eul Kilb e l. 4 becoe ecil ce of 9.4 : π Φ 9.5 FRCTIONL OPRTORS ND FUNCTION c Coide he fucio f whee Re > >! d c i biy co he he fciol iegl oeo of ode c be wie 5

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ I f c.. We deoe he fucio. c q i.e. c c.. The fciol diffeeil oeo of ode µ c be wie c µ µ D f D I! µ c..3 µ We deoe he fucio.3 c µ q i.e. THORM.. If Re > iegl oeo of ode he I D λ λ µ c c.4 µ µ > c i biy co d fciol c c λ..5 c c λ The Llce fo of c i give..6 c { c } L.7 I he ligh of Theoe. we c ove followig Theoe.. THORM.. If Re > iegl oeo of ode µ he I D λ λ > c i biy co d fciol c µ q c λ µ..8 c µ q c λ µ..9 c L{ c µ } µ.. 5

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ GNRTING RLTIONS ND FINIT SUMMTION FORMUL OF 6 coidebly lge ube of ecil fucio icludig ll of he clicl ohogol olyoil e ow o oe geeig elio. We ued oeiol echique by eloyig θ Mil 977 Pil d The 975 diffeeil oeo whee θ D d d θ D D fo obiig followig geeig elio d fiie d uio foule of.6. { } [ { }].. u e! ρ u. { } [ { { } { [ { }] }]..3 }..4 5

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ 53 } }] { [ } { {..5 Uig. o.5 we obied followig geeig elio }] { [ } { S ] [!..6 < N }] { [ } { S ] [!..7 < N Two fiie uio foule fo.6 lo obied

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ 54!..8! σ σ..9 we ge followig bilel geeig elio fo q b y R ] [ }] { [ } { b b y Φ. Whe Φ ] [ b b d y R b i olyoil of degee b i y which i defied y R b [ ] b y b / b i oiive iege i biy cole ube. RFRNCS. Cli L.: Soe eio d covoluio foul eled o Mc Moh e heoe SIM Joul of Mheicl lyi 8 3 336 977.. Goeflo R Kilb.. d Rogoi S.V.: O he geelied Mig-Leffle ye fucio Iegl Tfo d Secil Fucio 7 5--4 998. 3. Goeflo R. d Midi F.: O Mig Leffle fucio i fciol evluio ocee Joul of Couiol d lied Mheic 8 83 99.

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/ 4. Kh M.. d buhh G.S.: udy o wo vible logue of cei fciol oeo Po Mheic 733 3--48 3. 5. Kilb.. d Sigo M.: O Mig Leffle ye fucio fciol clculu oeo d oluio of Iegl equio Iegl Tfo d Secil Fucio 4 355 37 996. 6. Kilb.. Sigo M. d Se R.K.: Geelied Mig-Leffle fucio d geelied Fciol clculu oeo Iegl Tfo d Secil Fucio 5 3--49 4. 7. Mig Leffle G.M.: Su l ouvelle focio C.R. cd. Sci. Pi 37 554 558 93. 8. Mil H.B.: Bilie d bilel geeig elio eic Joul of Mheic 99 3--55 977. 9. Pil K.R. d The N.K.: Oeiol foul fo fucio defied by geelied Rodigue foul-ii Sciece Joul of Shivji Uiveiy 5 - - 975.. Pbh T.R.: igul iegl equio wih geelied Mig-Leffle fucio i he Keel Yooh Mheicl Joul 9 7--5 97.. Pji J.C.: Geeliio of Mig-Leffle fucio d i licio Ph.D. hei Dee of Mheic Sd Vllbhbhi Niol Iiue of Techology Su Idi 8.. Riville.D.: Secil Fucio The Mcill Coy New Yo 96. 3. Sigo M. d Kilb..: O Mig Leffle ye fucio d licio Iegl Tfo d Secil Fucio 7 97-- 998. 4. Seddo I.N.: The ue of Iegl Tfo T McGw Hill Pub. Co. Ld. New Delhi 979. 5. Shul.K. d Pji J.C.: O geeliio of Mig-Leffle fucio d i oeie Joul of Mheicl lyi d licio 337. 797--8 7. 6. Shul.K. d Pji J.C.: Soe oeie of cl of Polyoil uggeed by Mil Poyeccioe Joul of Mheic 6. 45-- 56 7 B. 7. Shul.K. d Pji J.C.: Geeliio of cl of Polyoil Deoio Mheic 4 4 89--86 7 C. 8. Shul K d Pji J.C.: geel cl of olyoil ocied wih geelied Mig-Leffle fucio. Iegl Tfo d Secil Fucio 9. 3--34 8. 9. Shul K d Pji J.C.: O Geelied Mig-Leffle ye fucio d geeed iegl oeo Mheicl Sciece Reech Joul. 83--9 8 B.. Shul.K. d Pji J.C.: Soe e o geelied Mig-Leffle fucio Poyeccioe Joul of Mheic 8 7--34 9.. Shul.K. d Pji J.C.: O Recuece elio of geelied Mig- Leffle fucio. Suvey i Mheic d i licio 4 33--38 9 B. 55

Ieiol Joul of Mheicl gieeig d Sciece ISSN : 77 698 Volue Iue h://ie.google.co/ie/ijejoul/ h://www.ije.co/. Shul.K. d Pji J.C.: Decooiio d oeie of geelied Mig-Leffle fucio dvce i Pue Mheic 8--4. 3. Sivv H.M. d Toovi Z.: Fciol clculu wih iegl oeo coiig geelied Mig-Leffle fucio i he eel lied Mheic d Couio 98 9. 4. Toovi Z. Hilfe R. d Sivv H.M.: Fciol d oeiol clculu wih geelied fciol deivive oeo d Mig-Leffle ye fucio Iegl Tfo d Secil Fucio 797--84. 5. Wi.: Ube de Fudel i de Theoie de Fuioe c Mheic 9 9 95. 56

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