On New Prajapati-Shukla Functions And Polynomials
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1 Mhei Toy Vol.7De ISSN O New Prji-Shul Fuio Polyoil J. C. Prji. K. Shul Dere of Mheil Siee Fuly of Tehology gieerig Chror Uiveriy of Siee Tehology Chg Ii. Dere of lie Mhei Huiie S.V.Niol Iiue of Tehology Sur Ii -il: jyoir8@reiffil.o jyhul@reiffil.o BSTRCT The riil i of he er i o iroue he vriou geerliio of Shul-Prji fuio olyoil. For hee ew fuio olyoil heir vriou roerie iluig uul iffereiio iegrio Iegrl rfor Geerlie hyergeoeri erie for Melli Bre iegrl rereeio Reurree relio Iegrl rereeio Deooiio Friol lulu oeror roerie geerig relio bilerl geerig relio fiie uio forule of ew l of olyoil lo eblihe. Key Wor: Mig Leffler fuio Iegrl Trfor Friol iegrl iffereil oeror. MS lifiio: 33 33C 44 3.
2 J. Prji &. Shul - O New Prji-Shul Fuio Polyoil 5. INTRODUCTION ND PRLIMINRIS Reely Shul Prji 7 iveige uie he fuio whih i efie for C Re Re Re N :. where! eoe he geerlie Pohher ybol Riville96 whih i riulr reue o r r if N. I oiuio of he uy he geerliio of be wrie whih Re Re Re i efie for!.. Thi i geerliio of he eoeil fuio e he oflue hyergeoeri fuio Riville 96 he Mig Leffler fuio Mig-Leffler 93 he Wi fuio Wi 95 he fuio efie by Prbhr 97 well euio.. Goreflo e l. 998 Goreflo Miri Kilb e l Sigo Kilb 998 Srivv Toovi 9 Toovi e l i re y oher reerher lo uie he vriou roerie of Mig-Leffler fuio i geerliio wih heir liio. The liio of Iegrl Trfor iue by Seo 979 The fuio i eire fuio of orer Re if Re boluely overge i R R if Re. The rue ower erie of he fuio be efie N N! N j j j.3 eil e for he uy of for : j j! j.4 where N Re Re.
3 6 Mhei Toy Vol.7De uhor iveige he oeror where w f w f w ] f..5 wc Re Re Re. If he.5 reue o he reul Prbhr 97. Shul Prji 8 iroue l of olyoil whih re oee by Mig-Leffler fuio i oiuio of he uy of l of olyoil uhor iroue geerl l of olyoil efie.6! where re rel or ole uber re o geerlie Mig- Leffler fuio efie.. The roof of ll reul eblihe i hi er re rllel o Shul Prji 7 7 B 7 C 8 8 B 9 9 B.. BSIC PROPORTIS OF TH FUNCTION oeuee of he efiiio. he followig reul hol: THORM.. If i riulr he C Re Re Re!....3 THORM.. If wc Re Re Re he for N.4 w Re.5
4 J. Prji &. Shul - O New Prji-Shul Fuio Polyoil 7 i riulr w.6 w w..7 THORM.3. If wih r N relively rie C he r i riulr r r r r e r r r! r r.8 r r = r 3. GNRLIZD HYPRGOMTRIC FUNCTION RPRSNTTION OF Uig. ig N N he we hve F. 3. Covergee rieri for geerlie hyergeoeri fuio F : i If he fuio F overge for ll fiie. ii If he fuio F overge for iverge for. iii If he fuio F iverge for. iv If he fuio F i boluely overge o he irle if Re j j i i.
5 8 Mhei Toy Vol.7De where i -ule i -ule MLLIN-BRNS INTGRL RPRSNTTION OF THORM 4.. Le R C rereee by he Melli Bre iegrl : N. The fuio i i 4. where L rg he oour of iegrio begiig i eig i iee o ere he ole of he iegr hoe for ll N o he righ. for ll N o he lef fro 5. INTGRL TRNSFORMS OF I hi eio oe ueful iegrl rfor lie uler rfor Lle rfor Melli rfor Whier rfor lo iue. THORM 5.. uler Be rfor b b b 5. where Re Re b Re Re Re Re. THORM 5.. Lle rfor. e 5.. where Re Re Re Re Re
6 J. Prji &. Shul - O New Prji-Shul Fuio Polyoil 9 THORM 5.3. Melli rfor. w w 5.3 where Re Re Re Re. To obi Whier rfor we ue he followig iegrl v v v W e v where Re v. 5.4 THORM 5.4. Whier Trfor. w W e 3 w 5.5 where Re Re Re Re Re. 6. RCURRNC RLTIONS THORM 6.. For y Re Re Re we ge where ] ]. THORM 6.. For N ] INTGRL RPRSNTTIONS THORM 7.. For y Re Re Re eig where N he = 7.
7 3 Mhei Toy Vol.7De THORM 7.. If Re Re Re Re he =. 7. Seil e of Theore 7.: For Re fro 7. he riulr e lie below: e = 7.3 oh = 7.4 ih = 7.5 e erf =. 7.6 THORM 7.3. If Re Re Re he = ]. 7.7 THORM 7.4. If Re Re Re he e = he THORM 7.5. If Re Re Re. 7.9 THORM 7.6. If Re Re Re he L. 7.
8 J. Prji &. Shul - O New Prji-Shul Fuio Polyoil 3 THORM 7.7. If Re Re Re he! e THORM 7.8. If Re Re Re Re Re Re C he u u u u If he w w w v v v v. 7.6 w w. 7.7 i riulr w w 7.8 w w. 7.9
9 3 Mhei Toy Vol.7De DCOMPOSITION OF MITTG-LFFLR FUNCTION THORM 8.. Iegrl Rereeio of he fuio = where + Re Re THORM 8.. If. Re Re u u u 8. he!! 8. THORM 8.3. Re 8.3 where Re Re. Rer of Theore 8.3: I i ey o verify h N N THORM 8.4. If Re Re he N T 8.5 where T N N N N N N N! N Re FRCTIONL INTGRL ND DIFFRNTIL OPRTORS SSOCITDWITH TH FUNCTION The followig well-ow f re rere for uyig roerie of he Rie- Liouville friol iegrl iffereil oeror oie wih he fuio lo he roerie of oeror f w.
10 J. Prji &. Shul - O New Prji-Shul Fuio Polyoil 33 L b Se of Lebegue eurble rel or ole vlue fuio Kilb e l.4: L b Coi of Lebegue eurble rel or ole vlue fuio f o b] i.e. L b = b f : f f. 9. Kilb e l. 4 uie he everl roerie of friol iegrl iffereil iegrl oeror. Coflue hyergeoeri fuio Riville 96: Thi i lo ow he Pohher Bre oflue hyergeoeri fuio efie b F b 9. b! where b or egive ieger i overge for ll fiie. Gu uliliio heore Riville 96: If i oiive ieger C he. 9.3 Rie-Liouville friol iegrl of orer Kh buhh GS 3 Le f L b C Re he I f f = I f = 9.4 i lle R-L lef-ie friol iegrl of orer. Le f L b C Re he I b f f = I b f = b b 9.5 i lle R-L righ-ie friol iegrl of orer.
11 34 Mhei Toy Vol.7De Theore 9.. Le R le w C Re Re Re Re for he w } I = w ] 9.6 w } D = w ]. 9.7 Theore 9.. Le C he I ] = 9.8 i riulr I e. 9.9 Theore 9.3. Le R le w C Re Re Re Re for he w = w. 9. Theore 9.4. Le R le w C Re Re Re Re forb he he oeror w i boue o L b w f B f 9. where B b Re Re Re ] w b! Re 9. he relio I w f w f 9.3 hol for y uble fuio f L b. We ere he fuio N :. 9.4!
12 J. Prji &. Shul - O New Prji-Shul Fuio Polyoil 35 O ubiuig i 9.4 he he reul Kilb e l. 4 beoe eil e of 9.4 : FRCTIONL OPRTORS ND FUNCTION Coier he fuio! f where Re i rbirry o he he friol iegrl oeror of orer be wrie f I.. We eoe he fuio. i.e... The friol iffereil oeror of orer be wrie! I D f D..3 We eoe he fuio.3 i.e..4 THORM.. If Re i rbirry o friol iegrl oeror of orer he I..5 D..6 The Lle rfor of i give } L.7 I he ligh of Theore. we rove followig Theore.. THORM.. If Re i rbirry o friol iegrl oeror of orer he I..8
13 36 Mhei Toy Vol.7De D..9 } L... GNRTING RLTIONS ND FINIT SUMMTION FORMUL OF 6 oierbly lrge uber of eil fuio iluig ll of he lil orhogol olyoil re ow o oe geerig relio. We ue oeriol ehiue by eloyig Mil 977 Pil Thre 975 iffereil oeror where D D D for obiig followig geerig relio fiie uio forule of.6. = }] }..! u u e.. = }] }..3 = } }] }..4 = } }] }..5
14 J. Prji &. Shul - O New Prji-Shul Fuio Polyoil 37 Uig. o.5 we obie followig geerig relio = }] } S ]!.6 N = }] } S ]!..7 N Two fiie uio forule for.6 lo obie!..8!..9 we ge followig bilerl geerig relio for b y R = ] }] } b b y. where ] b b
15 38 Mhei Toy Vol.7De i olyoil of egree R b y b i y whih i efie R b = y / b b y b i oiive ieger i rbirry ole uber. RFRNCS ] Crli L. 977: Soe eio ovoluio forul rele o M Moh er heore SIM Jourl of Mheil lyi ] Goreflo R Kilb.. Rogoi S.V. 998: O he geerlie Mig-Leffler ye fuio Iegrl Trfor Seil Fuio ] Goreflo R. Miri F. : O Mig Leffler fuio i friol evluio roee Jourl of Couiol lie Mhei ] Kh M.. buhh G.S. 3: uy o wo vrible logue of eri friol oeror Pro Mhei ] Kilb.. Sigo M. 996: O Mig Leffler ye fuio friol lulu oeror oluio of Iegrl euio Iegrl Trfor Seil Fuio ] Kilb.. Sigo M. Se R.K. 4: Geerlie Mig-Leffler fuio geerlie Friol lulu oeror Iegrl Trfor Seil Fuio ] Mig Leffler G.M. 93 : Sur l ouvelle foio C.R.. Si. Pri ] Mil H.B. 977: Bilier bilerl geerig relio eri Jourl of Mhei ] Pil K.R. Thre N.K. 975: Oeriol forul for fuio efie by geerlie Rorigue forul-ii Siee Jourl of Shivji Uiveriy 5 -. ] Prbhr T.R. 97: igulr iegrl euio wih geerlie Mig-Leffler fuio i he Kerel Yooh Mheil Jourl ] Prji J.C. 8. Geerliio of Mig-Leffler fuio i liio Ph.D. hei Dere of Mhei Srr Vllbhbhi Niol Iiue of Tehology Sur Ii ] Riville.D. 96: Seil Fuio The Mill Coy New Yor.
16 J. Prji &. Shul - O New Prji-Shul Fuio Polyoil 39 3] Sigo M. Kilb O Mig Leffler ye fuio liio Iegrl Trfor Seil Fuio ] Seo I.N. 979: The ue of Iegrl Trfor T MGrw Hill Pub. Co. L. New Delhi. 5] Shul.K. Prji J.C. 7 : O geerliio of Mig-Leffler fuio i roerie Jourl of Mheil lyi liio ] Shul.K. Prji J.C. 7 B: Soe roerie of l of Polyoil uggee by Mil Proyeioe Jourl of Mhei ] Shul.K. Prji J.C. 7 C. Geerliio of l of Polyoil Deorio Mhei ] Shul K Prji JC 8. geerl l of olyoil oie wih geerlie Mig-Leffler fuio. Iegrl Trfor Seil Fuio ] Shul K Prji JC 8 B. O Geerlie Mig-Leffler ye fuio geere iegrl oeror Mheil Siee Reerh Jourl ] Shul.K. Prji J.C. 9 : Soe rer o geerlie Mig-Leffler fuio Proyeioe Jourl of Mhei ] Shul.K. Prji J.C. 9 B: O Reurree relio of geerlie Mig-Leffler fuio. Survey i Mhei i liio ] Shul.K. Prji J.C. : Deooiio roerie of geerlie Mig-Leffler fuio Couie for ubliio. 3] Srivv H.M. Toovi Z. 9: Friol lulu wih iegrl oeror oiig geerlie Mig-Leffler fuio i he erel lie Mhei Couio 98. 4] Toovi Z. Hilfer R. Srivv H.M. : Friol oeriol lulu wih geerlie friol erivive oeror Mig-Leffler ye fuio Iegrl Trfor Seil Fuio ] Wi. 95: Uber e Fuel i er Theorie er Fuioe Mhei 9 9.
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